0,2

The brown rat (rattus norwegicus) breeds very quickly. It can give birth to other rats 7 times a year, starting at the age of three months. The average number of pups is 8. The present sequence gives the total number of rats, when the intervals are 12/7 of a year and a young rat starts having offspring at 24/7 of a year. -Hans Isdahl,Jan 26 2008

Numbers n such that tau(n) is odd where tau(x) denotes the Ramanujan tau function (A000594). -Benoit Cloitre,May 01 2003

If Y is a fixed 2-subset of a (2n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. -Milan Janjic,Oct 21 2007

Binomial transform of [1, 8, 8, 0, 0, 0,...]; Narayana transform (A001263) of [1, 8, 0, 0, 0,...]. -Gary W. Adamson,Dec 29 2007

All terms of this sequence are of the form 8k+1. For numbers 8k+1 which aren't squares seeA138393.Numbers 8k+1 are squares iff k is a triangular number fromA000217.And squares have form 4n(n+1)+1. -Artur Jasinski,Mar 27 2008

Sequence arises from reading the line from 1, in the direction 1, 25,... and the line from 9, in the direction 9, 49,..., in the square spiral whose vertices are the squaresA000290.-Omar E. Pol,May 24 2008

Equals the triangular numbers convolved with [1, 6, 1, 0, 0, 0,...]. -Gary W. Adamson&Alexander R. Povolotsky,May 29 2009

First differences:A008590(n) = a(n) - a(n-1) for n>0. -Reinhard Zumkeller,Nov 08 2009

Central terms of the triangle inA176271;cf.A000466,A053755.-Reinhard Zumkeller,Apr 13 2010

Odd numbers with odd abundance. Odd numbers with even abundance are inA088828.Even numbers with odd abundance are inA088827.Even numbers with even abundance are inA088829.-Jaroslav Krizek,May 07 2011

Appear as numerators in the non-simple continued fraction expansion of Pi-3: Pi-3 = K_{k>=1} (1-2*k)^2/6 = 1/(6+9/(6+25/(6+49/(6+...)))), see also the comment inA007509.-Alexander R. Povolotsky,Oct 12 2011

Ulam's spiral (SE spoke). -Robert G. Wilson v,Oct 31 2011

All terms end in 1, 5 or 9. Modulo 100, all terms are among { 1, 9, 21, 25, 29, 41, 49, 61, 69, 81, 89 }. -M. F. Hasler,Mar 19 2012

Right edge of both trianglesA214604andA214661:a(n) =A214604(n+1,n+1) =A214661(n+1,n+1). -Reinhard Zumkeller,Jul 25 2012

Also: Odd numbers which have an odd sum of divisors (= sigma =A000203). -M. F. Hasler,Feb 23 2013

Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c (A020882) and respective even leg b (A231100); sequence gives values c-b, sorted with duplicates removed. -K. G. Stier,Nov 04 2013

For n>1 a(n) is twice the area of the irregular quadrilateral created by the points ((n-2)*(n-1),(n-1)*n/2), ((n-1)*n/2,n*(n+1)/2), ((n+1)*(n+2)/2,n*(n+1)/2), and ((n+2)*(n+3)/2,(n+1)*(n+2)/2). -J. M. Bergot,May 27 2014

Number of pairs (x, y) of Z^2, such that max(abs(x), abs(y)) <= n. -Michel Marcus,Nov 28 2014

Except for a(1)=4, the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood. -Robert Price,May 23 2016

a(n) is the sum of 2n+1 consecutive numbers, the first of which is n+1. -Ivan N. Ianakiev,Dec 21 2016

a(n) is the number of 2 X 2 matrices with all elements in {0..n} with determinant = 2*permanent. -Indranil Ghosh,Dec 25 2016

Engel expansion of Pi*StruveL_0(1)/2 where StruveL_0(1) isA197037.-Benedict W. J. Irwin,Jun 21 2018

Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; the segments on the hypotenuse {p = a(n)/A001844(n), q =A060300(n)/A001844(n) =A001844(n) - p} and their ratio p/q = a(n)/A060300(n) are irreducible fractions in Q\Z. X values areA005408,Y values areA046092,Z values areA001844.-Ralf Steiner,Feb 25 2020

a(n) is the number of large or small squares that are used to tile primitive squares of type 2 (A344332). -Bernard Schott,Jun 03 2021

Also, positive odd integers with an odd number of odd divisors (for similar sequence with 'even', seeA348005). -Bernard Schott,Nov 21 2021

a(n) is the least odd number k = x + y, with 0 < x < y, such that there are n distinct pairs (x,y) for which x*y/k is an integer; for example, a(2) = 25 and the two corresponding pairs are (5,20) and (10,15). The similar sequence with 'even' isA016742(see Comment of Jan 26 2018). -Bernard Schott,Feb 24 2023

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 586.

Paolo Xausa,Table of n, a(n) for n = 0..9999(terms 0..1000 from T. D. Noe)

Jeremiah Bartz, Bruce Dearden, and Joel Iiams,Classes of Gap Balancing Numbers,arXiv:1810.07895 [math.NT], 2018.

Bruce C. Berndt and Ken Ono,Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary,Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.

John Elias,Illustration: 8-fold Square Progression of Ulam's Spiral.

Milan Janjic,Two Enumerative Functions;also onSemantic Scholar.

Scientific American,Cover of the March 1964 issue.

Amelia Carolina Sparavigna,Groupoids of OEIS A002378 and A016754 Numbers (oblong and odd square numbers),Politecnico di Torino (Italy, 2019).

Leo Tavares,Illustration: Diamond Triangles.

Leo Tavares,Illustration: Diamond Stars.

Eric Weisstein's World of Mathematics,Moore Neighborhood.

R. Yin, J. Mu, and T. Komatsu,The p-Frobenius Number for the Triple of the Generalized Star Numbers,Preprints 2024, 2024072280. See p. 2.

Index entries for sequences related to centered polygonal numbers.

Index entries for linear recurrences with constant coefficients,signature (3,-3,1).

a(n) = 1 + Sum_{i=1..n} 8*i = 1 + 8*A000217(n). - Xavier Acloque, Jan 21 2003;Zak Seidov,May 07 2006;Robert G. Wilson v,Dec 29 2010

O.g.f.: (1+6*x+x^2)/(1-x)^3. -R. J. Mathar,Jan 11 2008

a(n) = 4*n*(n + 1) + 1 = 4*n^2 + 4*n + 1. -Artur Jasinski,Mar 27 2008

a(n) =A061038(2+4n). -Paul Curtz,Oct 26 2008

Sum_{n>=0} 1/a(n) = Pi^2/8 =A111003.-Jaume Oliver Lafont,Mar 07 2009

a(n) =A000290(A005408(n)). -Reinhard Zumkeller,Nov 08 2009

a(n) = a(n-1) + 8*n with n>0, a(0)=1. -Vincenzo Librandi,Aug 01 2010

a(n) =A033951(n) + n. -Reinhard Zumkeller,May 17 2009

a(n) =A033996(n) + 1. -Omar E. Pol,Oct 03 2011

a(n) = (A005408(n))^2. -Zak Seidov,Nov 29 2011

FromGeorge F. Johnson,Sep 05 2012: (Start)

a(n+1) = a(n) + 4 + 4*sqrt(a(n)).

a(n-1) = a(n) + 4 - 4*sqrt(a(n)).

a(n+1) = 2*a(n) - a(n-1) + 8.

a(n+1) = 3*a(n) - 3*a(n-1) + a(n-2).

(a(n+1) - a(n-1))/8 = sqrt(a(n)).

a(n+1)*a(n-1) = (a(n)-4)^2.

a(n) = 2*A046092(n) + 1 = 2*A001844(n) - 1 =A046092(n) +A001844(n).

Limit_{n -> oo} a(n)/a(n-1) = 1. (End)

a(n) = binomial(2*n+2,2) + binomial(2*n+1,2). -John Molokach,Jul 12 2013

E.g.f.: (1 + 8*x + 4*x^2)*exp(x). -Ilya Gutkovskiy,May 23 2016

a(n) =A101321(8,n). -R. J. Mathar,Jul 28 2016

Product_{n>=1}A033996(n)/a(n) = Pi/4. -Daniel Suteu,Dec 25 2016

a(n) =A014105(n) +A000384(n+1). -Bruce J. Nicholson,Nov 11 2017

a(n) =A003215(n) +A002378(n). -Klaus Purath,Jun 09 2020

FromAmiram Eldar,Jun 20 2020: (Start)

Sum_{n>=0} a(n)/n! = 13*e.

Sum_{n>=0} (-1)^(n+1)*a(n)/n! = 3/e. (End)

Sum_{n>=0} (-1)^n/a(n) =A006752.-Amiram Eldar,Oct 10 2020

FromAmiram Eldar,Jan 28 2021: (Start)

Product_{n>=0} (1 + 1/a(n)) = cosh(Pi/2).

Product_{n>=1} (1 - 1/a(n)) = Pi/4 (A003881). (End)

FromLeo Tavares,Nov 24 2021: (Start)

a(n) =A014634(n) -A002943(n). See Diamond Triangles illustration.

a(n) =A003154(n+1) -A046092(n). See Diamond Stars illustration. (End)

FromPeter Bala,Mar 11 2024: (Start)

Sum_{k = 1..n+1} 1/(k*a(k)*a(k-1)) = 1/(9 - 3/(17 - 60/(33 - 315/(57 -... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*2^2 ))))).

3/2 - 2*log(2) = Sum_{k >= 1} 1/(k*a(k)*a(k-1)) = 1/(9 - 3/(17 - 60/(33 - 315/(57 -... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*2^2 -... ))))).

Row 2 ofA142992.(End)

FromPeter Bala,Mar 26 2024: (Start)

8*a(n) = (2*n + 1)*(a(n+1) - a(n-1)).

Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 1/2 - Pi/8 = 1/(9 + (1*3)/(8 + (3*5)/(8 +... + (4*n^2 - 1)/(8 +... )))). For the continued fraction use Lorentzen and Waadeland, p. 586, equation 4.7.9 with n = 1. Cf.A057813.(End)

A016754[nmax_]:=Range[1, 2nmax+1, 2]^2;A016754[100] (*Paolo Xausa,Mar 05 2023 *)

(PARI)A016754(n)=(n<<1+1)^2 \\Charles R Greathouse IV,Jun 16 2011, corrected and edited byM. F. Hasler,Apr 11 2023

(Haskell)

a016754 n = a016754_list!! n

a016754_list = scanl (+) 1 $ tail a008590_list

--Reinhard Zumkeller,Apr 02 2012

(Maxima)A016754(n):=(n+n+1)^2$

makelist(A016754(n), n, 0, 20); /*Martin Ettl,Nov 12 2012 */

(Magma) [n^2: n in [1..100 by 2]]; //Vincenzo Librandi,Jan 03 2017

(Python)

defA016754(n): return ((n<<1)|1)**2 #Chai Wah Wu,Jul 06 2023

Cf.A000290,A000384,A001263,A001539,A001844,A003881,A005408,A006752,A014105,A016742,A016802,A016814,A016826,A016838,A033996,A046092,A060300,A138393,A167661,A167700.

Cf.A000447(partial sums).

Cf.A005917,A344330,A344332.

Cf.A348005.

Partial sums ofA022144.

Sequences on the four axes of the square spiral: Starting at 0:A001107,A033991,A007742,A033954;starting at 1:A054552,A054556,A054567,A033951.

Sequences on the four diagonals of the square spiral: Starting at 0:A002939= 2*A000384,A016742= 4*A000290,A002943= 2*A014105,A033996= 8*A000217;starting at 1:A054554,A053755,A054569,A016754.

Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0:A035608,A156859,A002378= 2*A000217,A137932= 4*A002620;starting at 1:A317186,A267682,A002061,A080335.

Cf.A014634,A003154.

nonn,easy

Additional description fromTerrel Trotter, Jr.,Apr 06 2002

approved

0,2

Write the natural numbers in groups: 1; 2,3,4; 5,6,7,8,9; 10,11,12,13,14,15,16;..... and add the groups, i.e., a(n) = Sum_{j=n^2-2(n-1)..n^2} j. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Sep 05 2001

The numbers 1, 9, 35, 91, etc. are divisible by 1, 3, 5, 7, etc. Therefore there are no prime numbers in this list. 9 is divisible by 3 and every third number after 9 is also divisible by 3. 35 is divisible by 5 and 7 and every fifth number after 35 is also divisible by 5 and every seventh number after 35 is also divisible by 7. This pattern continues indefinitely. - Howard Berman (howard_berman(AT)hotmail.com), Nov 07 2008

n^3 + (n+1)^3 = (2n+1)*(n^2+n+1), hence all terms are composite. -Zak Seidov,Feb 08 2011

This is the order of an n-ball centered at a node in the Kronecker product (or direct product) of three cycles, each of whose lengths is at least 2n+2. -Pranava K. Jha,Oct 10 2011

Positive y values of 4*x^3 - 3*x^2 = y^2. -Bruno Berselli,Apr 28 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe,Table of n, a(n) for n = 0..1000

Pranava K. Jha,Perfect r-domination in the Kronecker product of three cycles,IEEE Trans. Circuits and Systems-I: Fundamental Theory and Applications, vol. 49, no. 1, pp. 89 - 92, Jan. 2002.

T. P. Martin,Shells of atoms,Phys. Reports, 273 (1996), 199-241, eq. (10).

Michael Penn,what's the pattern, Kenneth?,YouTube video, 2021.

Simon Plouffe,Approximations de séries génératrices et quelques conjectures,Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe,1031 Generating Functions,Appendix to Thesis, Montreal, 1992

B. K. Teo and N. J. A. Sloane,Magic numbers in polygonal and polyhedral clusters,Inorgan. Chem. 24 (1985), 4545-4558.

Eric Weisstein's World of Mathematics,Centered Cube Number

D. Zeitlin,A family of Galileo sequences,Amer. Math. Monthly 82 (1975), 819-822.

Index entries for linear recurrences with constant coefficients,signature (4,-6,4,-1).

a(n) = Sum_{i=0..n}A005897(i), partial sums. -Jonathan Vos Post,Feb 06 2011

G.f.: (x^2+4*x+1)*(1+x)/(1-x)^3. -Simon Plouffe(see MAPLE section) andColin Barker,Jan 02 2012; edited byN. J. A. Sloane,Feb 07 2018

a(n) =A037270(n+1) -A037270(n). -Ivan N. Ianakiev,May 13 2012

a(n) =A000217(n+1)^2 -A000217(n-1)^2. -Bob Selcoe,Mar 25 2016

a(n) =A005408(n) *A002061(n+1). -Miquel Cerda,Oct 05 2016

FromIlya Gutkovskiy,Oct 06 2016: (Start)

E.g.f.: (1 + 8*x + 9*x^2 + 2*x^3)*exp(x).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

a(n) = (A081435(n))^2 - (A081435(n) - 1)^2. -Sergey Pavlov,Mar 01 2017

A005898:=(z+1)*(z**2+4*z+1)/(z-1)**4; #Simon Plouffein his 1992 dissertation

a[n_]:=n^3; Table[a[n]+a[n+1], {n, 0, 100}] (*Vladimir Joseph Stephan Orlovsky,Jan 03 2009 *)

CoefficientList[Series[(1 + 5 x + 5 x^2 + x^3)/(1 - x)^4, {x, 0, 40}], x] (*Vincenzo Librandi,Dec 16 2015 *)

(Sage) [i^3+(i+1)^3 for i in range(0, 39)] #Zerinvary Lajos,Jul 03 2008

(Python)

A005898_list, m = [], [12, -6, 2, 1]

for _ in range(10**2):

A005898_list.append(m[-1])

for i in range(3):

m[i+1] += m[i] #Chai Wah Wu,Dec 15 2015

(Magma) [n^3+(n+1)^3: n in [0..40]]; //Vincenzo Librandi,Dec 16 2015

(PARI) a(n)=n^3 + (n+1)^3 \\Anders Hellström,Dec 16 2015

(1/12)*t*(2*n^3 - 3*n^2 + n) + 2*n - 1 for t = 2, 4, 6,... givesA049480,A005894,A063488,A001845,A063489,A005898,A063490,A057813,A063491,A005902,A063492,A005917,A063493,A063494,A063495,A063496.

Cf.A003215,A000537,A000578.-Vladimir Joseph Stephan Orlovsky,Jan 03 2009

Partial sums ofA005897.

The 28 uniform 3D tilings: cab:A299266,A299267;crs:A299268,A299269;fcu:A005901,A005902;fee:A299259,A299265;flu-e:A299272,A299273;fst:A299258,A299264;hal:A299274,A299275;hcp:A007899,A007202;hex:A005897,A005898;kag:A299256,A299262;lta:A008137,A299276;pcu:A005899,A001845;pcu-i:A299277,A299278;reo:A299279,A299280;reo-e:A299281,A299282;rho:A008137,A299276;sod:A005893,A005894;sve:A299255,A299261;svh:A299283,A299284;svj:A299254,A299260;svk:A010001,A063489;tca:A299285,A299286;tcd:A299287,A299288;tfs:A005899,A001845;tsi:A299289,A299290;ttw:A299257,A299263;ubt:A299291,A299292;bnn:A007899,A007202.See the Proserpio link inA299266for overview.

nonn,easy

approved

0,2

Number of points in simple cubic lattice at most n steps from origin.

If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then a(n-6) is equal to the number of 6-subsets of X intersecting each Y_i (i=1,2,3). -Milan Janjic,Aug 26 2007

Equals binomial transform of [1, 6, 12, 8, 0, 0, 0,...] where (1, 6, 12, 8) = row 3 of the Chebyshev triangleA013609.-Gary W. Adamson,Jul 19 2008

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 4, a(n-2) = -coeff(charpoly(A,x),x^(n-3)). -Milan Janjic,Jan 26 2010

a(n) =A005408(n) *A097080(n-1) / 3. -Reinhard Zumkeller,Dec 15 2013

a(n) = D(3,n) where D are the Delannoy numbers (A008288). As such, a(n) gives the number of grid paths from (0,0) to (3,n) using steps that move one unit north, east, or northeast. -David Eppstein,Sep 07 2014

The first comment above can be re-expressed and generalized as follows: a(n) is the number of points in Z^3 that are L1 (Manhattan) distance <= n from any given point. Equivalently, due to a symmetry that is easier to see in the Delannoy numbers array (A008288), as a special case ofDmitry Zaitsev's Dec 10 2015 comment onA008288,a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= 3 from any given point. -Shel Kaphan,Jan 02 2023

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe,Table of n, a(n) for n = 0..1000

Luciano Ancora,The Square Pyramidal Number and other figurate numbers,ch. 4.

Bela Bajnok,Additive Combinatorics: A Menu of Research Problems,arXiv:1705.07444 [math.NT], May 2017. See Section 2.3.

D. Bump, K. Choi, P. Kurlberg, and J. Vaaler,A local Riemann hypothesis, Ipages 16 and 17

J. H. Conway and N. J. A. Sloane,Low-Dimensional Lattices VII: Coordination Sequences,Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).

Milan Janjic,Two Enumerative Functions

Milan Janjic,Hessenberg Matrices and Integer Sequences,J. Int. Seq. 13 (2010) # 10.7.8.

Milan Janjić,On Restricted Ternary Words and Insets,arXiv:1905.04465 [math.CO], 2019.

G. Kreweras,Sur les hiérarchies de segments,Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.

G. Kreweras,Sur les hiérarchies de segments,Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

T. P. Martin,Shells of atoms,Phys. Reports, 273 (1996), 199-241, eq. (10).

Simon Plouffe,Approximations de séries génératrices et quelques conjectures,Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe,1031 Generating Functions,Appendix to Thesis, Montreal, 1992

R. G. Stanton and D. D. Cowan,Note on a "square" functional equation,SIAM Rev., 12 (1970), 277-279.

Eric Weisstein's World of Mathematics,Haüy Construction

Eric Weisstein's World of Mathematics,Octahedral Number

Index entries for linear recurrences with constant coefficients,signature (4,-6,4,-1).

G.f.: (1+x)^3 /(1-x)^4. [conjectured (correctly) bySimon Plouffein his 1992 dissertation]

a(n) = (2*n+1)*(2*n^2 + 2*n + 3)/3.

First differences ofA014820(n). -Alexander Adamchuk,May 23 2006

a(n) = a(n-1) + 4*n^2 + 2, a(0)=1. -Vincenzo Librandi,Mar 27 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=1, a(1)=7, a(2)=25, a(3)=63. -Harvey P. Dale,Jun 05 2013

a(n) = Sum_{k=0..min(3,n)} 2^k * binomial(3,k) * binomial(n,k). See Bump et al. -Tom Copeland,Sep 05 2014

FromLuciano Ancora,Jan 08 2015: (Start)

a(n) = 2 *A000330(n) +A000330(n+1) +A000330(n-1).

a(n) =A005900(n) +A005900(n+1).

a(n) =A005900(n) +A000330(n) +A000330(n+1).

a(n) =A000330(n-1) +A000330(n) +A005900(n+1). (End)

a(n) =A002412(n+1) +A016061(n-1) for n > 0. -Bruce J. Nicholson,Nov 12 2017

E.g.f.: exp(x)*(3 + 18*x + 18*x^2 + 4*x^3)/3. -Stefano Spezia,Mar 14 2024

Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = 5/6 - log(2) = (1 - 1/2 + 1/3) - log(2). -Peter Bala,Mar 21 2024

Table[(4 n^3 - 6 n^2 + 8 n - 3)/3, {n, 100}] (*Vladimir Joseph Stephan Orlovsky,Jan 15 2011 *)

LinearRecurrence[{4, -6, 4, -1}, {1, 7, 25, 63}, 40] (*Harvey P. Dale,Jun 05 2013 *)

CoefficientList[Series[(1 + x)^3/(-1 + x)^4, {x, 0, 20}], x] (*Eric W. Weisstein,Sep 27 2017 *)

(PARI) a(n)=(2*n+1)*(2*n^2+2*n+3)/3 \\Charles R Greathouse IV,Dec 06 2011

(Haskell)

a001845 n = (2 * n + 1) * (2 * n ^ 2 + 2 * n + 3) `div` 3

--Reinhard Zumkeller,Dec 15 2013

Sums of 2 consecutive terms giveA008412.

(1/12)*t*(2*n^3 - 3*n^2 + n) + 2*n - 1 for t = 2, 4, 6,... givesA049480,A005894,A063488,A001845,A063489,A005898,A063490,A057813,A063491,A005902,A063492,A005917,A063493,A063494,A063495,A063496.

Partial sums ofA005899.

Cf.A001846,A001847,A001848,etc.,A014820,A013609.

Cf. alsoA240876,A002412,A016061,A005899.

The 28 uniform 3D tilings: cab:A299266,A299267;crs:A299268,A299269;fcu:A005901,A005902;fee:A299259,A299265;flu-e:A299272,A299273;fst:A299258,A299264;hal:A299274,A299275;hcp:A007899,A007202;hex:A005897,A005898;kag:A299256,A299262;lta:A008137,A299276;pcu:A005899,A001845;pcu-i:A299277,A299278;reo:A299279,A299280;reo-e:A299281,A299282;rho:A008137,A299276;sod:A005893,A005894;sve:A299255,A299261;svh:A299283,A299284;svj:A299254,A299260;svk:A010001,A063489;tca:A299285,A299286;tcd:A299287,A299288;tfs:A005899,A001845;tsi:A299289,A299290;ttw:A299257,A299263;ubt:A299291,A299292;bnn:A007899,A007202.See the Proserpio link inA299266for overview.

Row/column 3 ofA008288.

nonn,easy,nice

approved

0,2

Called "magic numbers" in some chemical contexts.

Partial sums ofA005901(n). -Lekraj Beedassy,Oct 30 2003

Equals binomial transform of [1, 12, 30, 20, 0, 0, 0,...]. -Gary W. Adamson,Aug 01 2008

Crystal ball sequence for A_3 lattice. -Michael Somos,Jun 03 2012

H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe,Table of n, a(n) for n = 0..1000

S. Bjornholm,Clusters, condensed matter in embryonic form,Contemp. Phys. 31 1990 pp. 309-324.

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).

Nicolas Gastineau, Olivier Togni,Coloring of the d-th power of the face-centered cubic grid,arXiv:1806.08136 [cs.DM], 2018.

D. R. Herrick,Home Page(displays these numbers as sizes of clusters in chemistry)

Xiaogang Liang, Ilyar Hamid, and Haiming Duan,Dynamic stabilities of icosahedral-like clusters and their ability to form quasicrystals,>, AIP Advances 6, 065017 (2016).

T. P. Martin,Shells of atoms,Phys. Reports, 273 (1996), 199-241, eq. (11).

Simon Plouffe,Approximations de séries génératrices et quelques conjectures,Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe,1031 Generating Functions,Appendix to Thesis, Montreal, 1992

B. K. Teo and N. J. A. Sloane,Magic numbers in polygonal and polyhedral clusters,Inorgan. Chem. 24 (1985), 4545-4558.

K. Urner,Cuboctahedral Sphere Packing

Index entries for linear recurrences with constant coefficients,signature (4,-6,4,-1).

a(n) = (2*n+1)*(5*n^2+5*n+3)/3.

For n > 0, n*a(n) = (Sum_{i=0..n-1} a(i)) + 2*A005891(n)*A000217(n). -Bruno Berselli,Feb 02 2011

a(-1 - n) = -a(n). -Michael Somos,Jun 03 2012

FromIndranil Ghosh,Apr 08 2017: (Start)

G.f.: (x^3 + 9x^2 + 9x + 1)/(x - 1)^4.

E.g.f.: (1/3)*exp(x)*(10x^3 + 45x^2 + 36x + 3).

(End)

a(n) =A100171(n+1) -A008778(n-1) =A100174(n+1) -A000290(n) =A005917(n+1) -A006331(n) =A051673(n+1) +A000578(n). -Bruce J. Nicholson,Jul 05 2018

a(4) = 147 = (1, 3, 3, 1) dot (1, 12, 30, 20) = (1 + 36 + 90 + 20). -Gary W. Adamson,Aug 01 2008

G.f. = 1 + 13*x + 55*x^2 + 147*x^3 + 309*x^4 + 561*x^5 + 923*x^6 + 1415*x^7 +...

A005902:= n -> (2*n+1)*(5*n^2+5*n+3)/3;

A005902:=(z+1)*(z**2+8*z+1)/(z-1)**4; #Simon Plouffein his 1992 dissertation

f[n_]:= (2n + 1)(5n^2 + 5n + 3)/3; Array[f, 36, 0] (*Robert G. Wilson v,Feb 02 2011 *)

LinearRecurrence[{4, -6, 4, -1}, {1, 13, 55, 147}, 50] (*Harvey P. Dale,Oct 08 2015 *)

CoefficientList[Series[(x^3 + 9*x^2 + 9*x + 1)/(x - 1)^4, {x, 0, 50}], x] (*Indranil Ghosh,Apr 08 2017 *)

(PARI) {a(n) = (2*n + 1) * (5*n^2 + 5*n + 3) / 3}; /*Michael Somos,Jun 03 2012 */

(PARI) x='x+O('x^50); Vec((x^3 + 9*x^2 + 9*x + 1)/(x - 1)^4) \\Indranil Ghosh,Apr 08 2017

(Magma) [(2*n+1)*(5*n^2+5*n+3)/3: n in [0..30]]; //G. C. Greubel,Dec 01 2017

(Python)

def a(n): return (2*n+1)*(5*n**2+5*n+3)//3

print([a(n) for n in range(40)]) #Michael S. Branicky,Jan 13 2021

(1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6,... givesA049480,A005894,A063488,A001845,A063489,A005898,A063490,A057813,A063491,A005902,A063492,A005917,A063493,A063494,A063495,A063496.

The 28 uniform 3D tilings: cab:A299266,A299267;crs:A299268,A299269;fcu:A005901,A005902;fee:A299259,A299265;flu-e:A299272,A299273;fst:A299258,A299264;hal:A299274,A299275;hcp:A007899,A007202;hex:A005897,A005898;kag:A299256,A299262;lta:A008137,A299276;pcu:A005899,A001845;pcu-i:A299277,A299278;reo:A299279,A299280;reo-e:A299281,A299282;rho:A008137,A299276;sod:A005893,A005894;sve:A299255,A299261;svh:A299283,A299284;svj:A299254,A299260;svk:A010001,A063489;tca:A299285,A299286;tcd:A299287,A299288;tfs:A005899,A001845;tsi:A299289,A299290;ttw:A299257,A299263;ubt:A299291,A299292;bnn:A007899,A007202.See the Proserpio link inA299266for overview.

Cf.A100171,A100174,A051673.

nonn,easy,nice

approved

0,2

Binomial transform of (1,4,6,4,0,0,0,...). -Paul Barry,Jul 01 2003

If X is an n-set and Y a fixed 4-subset of X then a(n-4) is equal to the number of 4-subsets of X intersecting Y. -Milan Janjic,Jul 30 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe,Table of n, a(n) for n = 0..1000

Milan Janjic,Two Enumerative Functions

T. P. Martin,Shells of atoms,Phys. Rep., 273 (1996), 199-241, eq. (10).

Simon Plouffe,Approximations de séries génératrices et quelques conjectures,Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe,1031 Generating Functions,Appendix to Thesis, Montreal, 1992

Luis Manuel Rivera,Integer sequences and k-commuting permutations,arXiv preprint arXiv:1406.3081 [math.CO], 2014.

B. K. Teo and N. J. A. Sloane,Magic numbers in polygonal and polyhedral clusters,Inorgan. Chem. 24 (1985), 4545-4558.

Index entries for linear recurrences with constant coefficients,signature (4, -6, 4, -1).

a(n) = (2*n + 1)*(n^2 + n + 3)/3.

G.f.: (1+x)*(1+x^2)/(1-x)^4.

a(n) = C(n, 0) + 4*C(n, 1) + 6*C(n, 2) + 4*C(n, 3). -Paul Barry,Jul 01 2003

a(n) is the sum of 4 consecutive tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)*(n+2)*(n+3)/6 =A000292(n). a(n) =A000292(n-3) +A000292(n-2) +A000292(n-1) +A000292(n). -Alexander Adamchuk,May 20 2006

a(n) = binomial(n+3,n) + binomial(n+2,n-1) + binomial(n+1,n-2) + binomial(n,n-3). (modified byG. C. Greubel,Nov 30 2017)

a(n) = a(n-1) + 2*n^2 + 2, n>=1 (first differencesA005893). -Vincenzo Librandi,Mar 27 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=5, a(2)=15, a(3)=35. -Harvey P. Dale,Nov 03 2011

E.g.f.: (3 + 12*x + 9*x^2 + 2*x^3)*exp(x)/3. -G. C. Greubel,Nov 30 2017

A005894:=(z+1)*(1+z**2)/(z-1)**4; #Simon Plouffein his 1992 dissertation

Table[(2n+1)(n^2+n+3)/3, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 5, 15, 35}, 40] (*Harvey P. Dale,Nov 03 2011 *)

(PARI) a(n)=(2*n+1)*(n^2+n+3)/3 \\Charles R Greathouse IV,Sep 24 2015

(Magma) [(2*n+1)*(n^2+n+3)/3: n in [0..30]]; //G. C. Greubel,Nov 30 2017

(1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6,... givesA049480,A005894,A063488,A001845,A063489,A005898,A063490,A057813,A063491,A005902,A063492,A005917,A063493,A063494,A063495,A063496.

Cf.A000292.

The 28 uniform 3D tilings: cab:A299266,A299267;crs:A299268,A299269;fcu:A005901,A005902;fee:A299259,A299265;flu-e:A299272,A299273;fst:A299258,A299264;hal:A299274,A299275;hcp:A007899,A007202;hex:A005897,A005898;kag:A299256,A299262;lta:A008137,A299276;pcu:A005899,A001845;pcu-i:A299277,A299278;reo:A299279,A299280;reo-e:A299281,A299282;rho:A008137,A299276;sod:A005893,A005894;sve:A299255,A299261;svh:A299283,A299284;svj:A299254,A299260;svk:A010001,A063489;tca:A299285,A299286;tcd:A299287,A299288;tfs:A005899,A001845;tsi:A299289,A299290;ttw:A299257,A299263;ubt:A299291,A299292;bnn:A007899,A007202.See the Proserpio link inA299266for overview.

nonn,easy,nice

approved

1,2

Harry J. Smith,Table of n, a(n) for n = 1..1000

T. P. Martin,Shells of atoms,Phys. Rep., 273 (1996), 199-241, eq. (10).

Index entries for linear recurrences with constant coefficients,signature (4, -6, 4, -1).

G.f.: x*(1+x)*(1+3*x+x^2)/(1-x)^4. -Colin Barker,Mar 02 2012

a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4), with a(1)=1, a(2)=8, a(3)=30, a(4)=77. -Harvey P. Dale,Aug 20 2012

E.g.f.: (-6 + 12*x + 15*x^2 + 10*x^3)*exp(x)/6 + 1. -G. C. Greubel,Dec 01 2017

Table[(2n-1)(5n^2-5n+6)/6, {n, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 8, 30, 77}, 40] (*Harvey P. Dale,Aug 20 2012 *)

(PARI) { for (n=1, 1000, write( "b063489.txt", n, "", (2*n - 1)*(5*n^2 - 5*n + 6)/6) ) } \\Harry J. Smith,Aug 23 2009

(Magma) [(2*n-1)*(5*n^2-5*n+6)/6: n in [1..30]]; //G. C. Greubel,Dec 01 2017

(PARI) x='x+O('x^30); Vec(serlaplace((-6 + 12*x + 15*x^2 + 10*x^3 )*exp(x)/6 + 1)) \\G. C. Greubel,Dec 01 2017

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6,... givesA049480,A005894,A063488,A001845,A063489,A005898,A063490,A057813,A063491,A005902,A063492,A005917,A063493,A063494,A063495,A063496.

Partial sums ofA010001.

The 28 uniform 3D tilings: cab:A299266,A299267;crs:A299268,A299269;fcu:A005901,A005902;fee:A299259,A299265;flu-e:A299272,A299273;fst:A299258,A299264;hal:A299274,A299275;hcp:A007899,A007202;hex:A005897,A005898;kag:A299256,A299262;lta:A008137,A299276;pcu:A005899,A001845;pcu-i:A299277,A299278;reo:A299279,A299280;reo-e:A299281,A299282;rho:A008137,A299276;sod:A005893,A005894;sve:A299255,A299261;svh:A299283,A299284;svj:A299254,A299260;svk:A010001,A063489;tca:A299285,A299286;tcd:A299287,A299288;tfs:A005899,A001845;tsi:A299289,A299290;ttw:A299257,A299263;ubt:A299291,A299292;bnn:A007899,A007202.See the Proserpio link inA299266for overview.

nonn,easy

N. J. A. Sloane,Aug 01 2001

approved

1,2

Final digits of a(n), i.e., a(n) mod 10, are repeated periodically with period of length 5 {1,5,5,5,9}. There is a symmetry in this list since the sum of two numbers equally distant from the ends is equal to 10 = 1 + 9 = 5 + 5 = 2*5. Last two digits of a(n), i.e., a(n) mod 100, are repeated periodically with period of length 50. -Alexander Adamchuk,Aug 11 2006

a(n) = VarScheme(n,2) in the scheme displayed inA128195.-Peter Luschny,Feb 26 2007

If Y is a 3-subset of a 2n-set X then, for n >= 2, a(n-2) is the number of 4-subsets of X intersecting Y. -Milan Janjic,Nov 18 2007

The numbers are the constant number found in magic squares of order n, where n is an odd number, see the comment inA006003.A Magic Square of side 1 is 1; 3 is 15; 5 is 65 and so on. -David Quentin Dauthier,Nov 07 2008

Two times the area of the triangle with vertices at (0,0), ((n - 1)^2, n^2), and (n^2, (n - 1)^2). -J. M. Bergot,Jun 25 2013

Bisection ofA006003.-Omar E. Pol,Sep 01 2018

Construct an array M with M(0,n) = 2*n^2 + 4*n + 1 =A056220(n+1), M(n,0) = 2*n^2 + 1 =A058331(n) and M(n,n) = 2*n*(n+1) + 1 =A001844(n). Row(n) begins with all the increasing odd numbers fromA058331(n) toA001844(n) and column(n) begins with all the decreasing odd numbers fromA056220(n+1) toA001844(n). The sum of the terms in row(n) plus those in column(n) minus M(n,n) equals a(n+1). The first five rows of array M are [1, 7, 17, 31, 49,...]; [3, 5, 15, 29, 47,...]; [9, 11, 13, 27, 45,...]; [19, 21, 23, 25, 43,...]; [33, 35, 37, 39, 41,...]. -J. M. Bergot,Jul 16 2013 [This contribution was moved here fromA047926byPetros Hadjicostas,Mar 08 2021.]

For n>=2, these are the primitive sides s of squares of type 2 described inA344332.-Bernard Schott,Jun 04 2021

(a(n) + 1) / 2 =A212133(n) is the number of cells in the n-th rhombic-dodecahedral polycube. -George Sicherman,Jan 21 2024

J. H. Conway and R. K. Guy, The Book of Numbers, p. 53.

E. Deza and M. M. Deza, Figurate Numbers, World Scientific Publishing, 2012, pp. 123-124.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi,Table of n, a(n) for n = 1..10000

Mario Defranco and Paul E. Gunnells,Hypergraph matrix models and generating functions,arXiv:2204.11361 [math.CO], 2022.

Milan Janjic,Two Enumerative Functions

T. P. Martin,Shells of atoms,Phys. Rep., 273 (1996), 199-241, eq. (9).

Andy Nicol,Illustration of Rhombic Dodecahedral Numbers

C. J. Pita Ruiz V.,Some Number Arrays Related to Pascal and Lucas Triangles,J. Int. Seq. 16 (2013) #13.5.7

Simon Plouffe,Approximations de séries génératrices et quelques conjectures,Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe,1031 Generating Functions,Appendix to Thesis, Montreal, 1992

B. K. Teo and N. J. A. Sloane,Magic numbers in polygonal and polyhedral clusters,Inorgan. Chem. 24 (1985), 4545-4558.

Eric Weisstein's World of Mathematics,Rhombic Dodecahedral Number.

Eric Weisstein's World of Mathematics,Nexus Number.

D. Zeitlin,A family of Galileo sequences,Amer. Math. Monthly 82 (1975), 819-822.

Index entries for linear recurrences with constant coefficients,signature (4,-6,4,-1).

a(n) = (2*n - 1)*(2*n^2 - 2*n + 1).

Sum_{i=1..n} a(i) = n^4 =A000583(n). First differences ofA000583.

G.f.: x*(1+x)*(1+10*x+x^2)/(1-x)^4. -Simon Plouffein his 1992 dissertation

More generally, g.f. for n^m - (n - 1)^m is Euler(m, x)/(1 - x)^m, where Euler(m, x) is Eulerian polynomial of degree m (cf.A008292). E.g.f.: x*(exp(y/(1 - x)) - exp(x*y/(1 - x)))/(exp(x*y/(1 - x))-x*exp(y/(1 - x))). -Vladeta Jovovic,May 08 2002

a(n) = sum of the next (2*n - 1) odd numbers; i.e., group the odd numbers so that the n-th group contains (2*n - 1) elements like this: (1), (3, 5, 7), (9, 11, 13, 15, 17), (19, 21, 23, 25, 27, 29, 31),... E.g., a(3) = 65 because 9 + 11 + 13 + 15 + 17 = 65. - Xavier Acloque, Oct 11 2003

a(n) = 2*n - 1 + 12*Sum_{i = 1..n} (i - 1)^2. - Xavier Acloque, Oct 16 2003

a(n) = (4*binomial(n,2) + 1)*sqrt(8*binomial(n,2) + 1). -Paul Barry,Mar 14 2004

Binomial transform of [1, 14, 36, 24, 0, 0, 0,...], if the offset is 0. -Gary W. Adamson,Dec 20 2007

Sum_{i=1..n-1}(a(i) + a(i+1)) = 8*Sum_{i=1..n}(i^3 + i) = 16*A002817(n-1) for n > 1. -Bruno Berselli,Mar 04 2011

a(n+1) = a(n) + 2*(6*n^2 + 1) = a(n) +A005914(n). -Vincenzo Librandi,Mar 16 2011

a(n) = -a(-n+1). a(n) = (1/6)*(A181475(n) -A181475(n-2)). -Bruno Berselli,Sep 26 2011

a(n) =A045975(2*n-1,n) =A204558(2*n-1)/(2*n - 1). -Reinhard Zumkeller,Jan 18 2012

a(n+1) = Sum_{k=0..2*n+1} (A176850(n,k) -A176850(n-1,k))*(2*k + 1), n >= 1. -L. Edson Jeffery,Nov 02 2012

a(n) =A005408(n-1) *A001844(n-1) = (2*(n - 1) + 1) * (2*(n - 1)*n + 1) =A000290(n-1)*12 + 2 + a(n-1). -Bruce J. Nicholson,May 17 2017

a(n) =A007588(n) +A007588(n-1) =A000292(2n-1) +A000292(2n-2) +A000292(2n-3) =A002817(2n-1) -A002817(2n-2). -Bruce J. Nicholson,Oct 22 2017

a(n) =A005898(n-1) + 6*A000330(n-1) (cf. Deza, Deza, 2012, p. 123, Section 2.6.2). -Felix Fröhlich,Oct 01 2018

a(n) =A300758(n-1) +A005408(n-1). -Bruce J. Nicholson,Apr 23 2020

G.f.: polylog(-4, x)*(1-x)/x. See theSimon Plouffeformula above (with expanded numerator), and the g.f. of the rows ofA008292byVladeta Jovovic,Sep 02 2002. -Wolfdieter Lang,May 10 2021

Table[n^4-(n-1)^4, {n, 40}] (*Harvey P. Dale,Apr 01 2011 *)

#[[2]]-#[[1]]&/@Partition[Range[0, 40]^4, 2, 1] (* More efficient than the above Mathematica program because it only has to calculate each 4th power once *) (*Harvey P. Dale,Feb 07 2015 *)

Differences[Range[0, 40]^4] (*Harvey P. Dale,Aug 11 2023 *)

(PARI) a(n)=n^4-(n-1)^4 \\Charles R Greathouse IV,Jul 31 2011

(Magma) [n^4 - (n-1)^4: n in [1..50]]; //Vincenzo Librandi,Aug 01 2011

(Haskell)

a005917 n = a005917_list!! (n-1)

a005917_list = map sum $ f 1 [1, 3..] where

f x ws = us: f (x + 2) vs where (us, vs) = splitAt x ws

--Reinhard Zumkeller,Nov 13 2014

(Python)

A005917_list, m = [], [24, -12, 2, 1]

for _ in range(10**2):

A005917_list.append(m[-1])

for i in range(3):

m[i+1] += m[i] #Chai Wah Wu,Dec 15 2015

(1/12)*t*(2*n^3 - 3*n^2 + n) + 2*n - 1 for t = 2, 4, 6,... givesA049480,A005894,A063488,A001845,A063489,A005898,A063490,A057813,A063491,A005902,A063492,A063493,A063494,A063495,A063496.

Column k=3 ofA047969.

Cf.A128195,A176850,A005408,A176271,A212133.

Cf.A001844,A000583,A000290.

Cf.A007588,A000292,A000332,A002817,A342354.

Cf.A031215,A008292.

Cf.A016754,A344330,A344332.

nonn,easy,nice

approved

1,2

Number of potential flows in a 2 X 2 matrix with integer velocities in -n..n, i.e., number of 2 X 2 matrices with adjacent elements differing by no more than n, counting matrices differing by a constant only once. -R. H. Hardin,Feb 27 2002

Number of ordered quadruples (a,b,c,d), -(n-1) <= a,b,c,d <= n-1, such that a+b+c+d = 0. -Benoit Cloitre,Jun 14 2003

If Y and Z are 2-blocks of a (2n+1)-set X then a(n-1) is the number of 5-subsets of X intersecting both Y and Z. -Milan Janjic,Oct 28 2007

Equals binomial transform of [1, 18, 48, 32, 0, 0, 0,...]. -Gary W. Adamson,Jul 19 2008

Harry J. Smith,Table of n, a(n) for n = 1..1000

R. Bacher, P. de la Harpe and B. Venkov,Séries de croissance et séries d'Ehrhart associées aux réseaux de racines,C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

Milan Janjic,Two Enumerative Functions

T. P. Martin,Shells of atoms,Phys. Rep., 273 (1996), 199-241, eq. (10).

Index entries for linear recurrences with constant coefficients,signature (4,-6,4,-1).

FromPeter Bala,Jul 18 2008: (Start)

The following remarks about the C_3 lattice assume the sequence offset is 0.

Partial sums ofA010006.So this sequence is the crystal ball sequence for the C_3 lattice - row 3 ofA142992.The lattice C_3 consists of all integer lattice points v = (a,b,c) in Z^3 such that a + b + c is even, equipped with the taxicab type norm ||v|| = (1/2) * (|a| + |b| + |c|).

The crystal ball sequence of C_3 gives the number of lattice points v in C_3 with ||v|| <= n for n = 0,1,2,3,... [Bacher et al.].

For example, a(1) = 19 because the origin has norm 0 and the 18 lattice points in Z^3 of norm 1 (as defined above) are +-(2,0,0), +-(0,2,0), +-(0,0,2), +-(1,1,0), +-(1,0,1), +-(0,1,1), +-(1,-1,0), +-(1,0,-1) and +-(0,1,-1). These 18 vectors form a root system of type C_3.

O.g.f.: x*(1 + 15*x + 15*x^2 + x^3)/(1 - x)^4 = x/(1 - x) * T(3, (1 + x)/(1 - x)), where T(n, x) denotes the Chebyshev polynomial of the first kind.

2*log(2) = 4/3 + Sum_{n >= 1} 1/(n*a(n)*a(n+1)). (End)

a(n+1) = (1/Pi) * Integral_{x=0..Pi} (sin((n+1/2)*x)/sin(x/2))^4. -Yalcin Aktar,Nov 02 2011, corrected byR. J. Mathar,Dec 01 2011

FromG. C. Greubel,Dec 01 2017: (Start)

G.f.: x*(1 + 15*x + 15*x^2 + x^3)/(1 - x)^4.

E.g.f.: (-3 + 6*x + 24*x^2 + 16*x^3)*exp(x)/3 + 1. (End)

a(n) =A005900(2n-1). -Ivan N. Ianakiev,Mar 27 2022

FromPeter Bala,Mar 11 2024: (Start)

Sum_{k = 1..n+1} 1/(k*a(k)*a(k+1)) = 1/(19 - 3/(27 - 60/(43 - 315/(67 -... -n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*3^2))))).

E.g.f.: exp(x)*(1 + 18*x + 48*x^2/2! + 32*x^3/3!). Note that -T(6, i*sqrt(x)) = 1 + 18*x + 48*x^2 + 32*x^3, where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. SeeA008310.(End)

A063496:=n->(2*n-1)*(8*n^2-8*n+3)/3; seq(A063496(n), n=1..40); #Wesley Ivan Hurt,May 09 2014

Table[(2*n - 1)*(8*n^2 - 8*n + 3)/3, {n, 40}] (*Wesley Ivan Hurt,May 09 2014 *)

LinearRecurrence[{4, -6, 4, -1}, {1, 19, 85, 231}, 30] (*G. C. Greubel,Dec 01 2017 *)

(PARI) { for (n=1, 1000, write( "b063496.txt", n, "", (2*n - 1)*(8*n^2 - 8*n + 3)/3) ) } \\Harry J. Smith,Aug 23 2009

(PARI) x='x+O('x^30); Vec(serlaplace((-3+6*x+24*x^2+16*x^3)*exp(x)/3 + 1)) \\G. C. Greubel,Dec 01 2017

(Magma) [(2*n-1)*(8*n^2-8*n+3)/3: n in [1..40]]; //Wesley Ivan Hurt,May 09 2014

(1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6,... givesA049480,A005894,A063488,A001845,A063489,A005898,A063490,A057813,A063491,A005902,A063492,A005917,A063493,A063494,A063495,A063496.

Cf.A003215,A010006,A142992,A142993,A142994.

nonn,easy

N. J. A. Sloane,Aug 01 2001

approved

1,2

FromOmar E. Pol,Oct 23 2019: (Start)

a(n) is also the sum of terms that are in the n-th finite row and in the n-th finite column of the square [1,n]x[1,n] of the natural number arrayA000027;e.g., the [1,3]x[1,3] square is

1..3..6

2..5..9

4..8..13,

so that a(1) = 1, a(2) = 2 + 3 + 5 = 10, a(3) = 4 + 6 + 8 + 9 + 13 = 40.

Hence the partial sums giveA185505.(End)

Harry J. Smith,Table of n, a(n) for n = 1..1000

Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn,Combinatorics on bounded free Motzkin paths and its applications,arXiv:2205.15554 [math.CO], 2022. (See p. 14).

T. P. Martin,Shells of atoms,Phys. Rep., 273 (1996), 199-241, eq. (10).

Index entries for linear recurrences with constant coefficients,signature (4,-6,4,-1).

G.f.: x*(1+x)*(1+5*x+x^2)/(1-x)^4. -Colin Barker,Mar 02 2012

a(n) = Sum_{k = n^2-2*n+2..n^2}A064788(k). -Lior Manor,Jan 13 2013

FromG. C. Greubel,Dec 01 2017: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

E.g.f.: (-6 + 12*x + 21*x^2 + 14*x^3)*exp(x)/6 + 1. (End)

Table[(2*n-1)*(7*n^2-7*n+6)/6, {n, 1, 50}] (* or *) LinearRecurrenc[{4, -6, 4, -1}, {1, 10, 40, 105}, 50] (*G. C. Greubel,Dec 01 2017 *)

(PARI) { for (n=1, 1000, write( "b063490.txt", n, "", (2*n - 1)*(7*n^2 - 7*n + 6)/6) ) } \\Harry J. Smith,Aug 23 2009

(PARI) x='x+O('x^30); Vec(serlaplace((-6 + 12*x + 21*x^2 + 14*x^3 )*exp(x)/6 + 1)) \\G. C. Greubel,Dec 01 2017

(Magma) [(2*n-1)*(7*n^2-7*n+6)/6: n in [1..30]]; //G. C. Greubel,Dec 01 2017

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6,... givesA049480,A005894,A063488,A001845,A063489,A005898,A063490,A057813,A063491,A005902,A063492,A005917,A063493,A063494,A063495,A063496.

Cf.A064788,A185505.

nonn,easy

N. J. A. Sloane,Aug 01 2001

approved

1,2

Sum of two consecutive terms ofA006003(n) = n*(n^2+1)/2. a(n) =A006003(n-1) +A006003(n). -Alexander Adamchuk,Jun 03 2006

If a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-4) is the number of 5-subsets of X intersecting both Y and Z. -Milan Janjic,Sep 08 2007

Harry J. Smith,Table of n, a(n) for n = 1..1000

Milan Janjic,Two Enumerative Functions

M. Janjic and B. Petkovic,A Counting Function,arXiv 1301.4550 [math.CO], 2013.

T. P. Martin,Shells of atoms,Phys. Rep., 273 (1996), 199-241, eq. (10).

Index entries for linear recurrences with constant coefficients,signature (4,-6,4,-1).

G.f.: (1 + x)*(1 + x + x^2)/(1 - x)^4. -Jaume Oliver Lafont,Aug 30 2009

a(n) =A000217(A000217(n)) -A000217(A000217(n-2)). -Bruno Berselli,Oct 14 2016

E.g.f.: (-2 + 4*x + 3*x^2 + 2*x^3)*exp(x)/2 + 1. -G. C. Greubel,Dec 01 2017

Table[(2 n - 1) (n^2 - n + 2)/2, {n, 1, 40}] (*Bruno Berselli,Oct 14 2016 *)

LinearRecurence[{4, -6, 4, -1}, {1, 6, 20, 49}, 50] (*G. C. Greubel,Dec 01 2017 *)

(PARI) { for (n=1, 1000, write( "b063488.txt", n, "", (2*n - 1)*(n^2 - n + 2)/2) ) } \\Harry J. Smith,Aug 23 2009

(PARI) x='x+O('x^30); Vec(serlaplace((-2 + 4*x + 3*x^2 + 2*x^3)*exp(x)/2 + 1)) \\G. C. Greubel,Dec 01 2017

(Magma) [(2*n-1)*(n^2 -n +2)/2: n in [1..30]]; //G. C. Greubel,Dec 01 2017

1/12*t*n*(2*n^2 - 3*n + 1) + 2*n - 1 for t = 2, 4, 6,... givesA049480,A005894,A063488,A001845,A063489,A005898,A063490,A057813,A063491,A005902,A063492,A005917,A063493,A063494,A063495,A063496.

Cf.A000217,A006003.

Partial sums ofA005918.

nonn,easy

N. J. A. Sloane,Aug 01 2001

approved