OFFSET
1,2
COMMENTS
Chebyshev S-sequence with Diophantine property.
4*b(n)^2 - 3*a(n)^2 = 1 with b(n) =A001570(n), n>=0.
y satisfying the Pellian x^2 - 3*y^2 = 1, for even x given byA094347(n). -Lekraj Beedassy,Jun 03 2004
a(n) = L(n,-14)*(-1)^n, where L is defined as inA108299;see alsoA001570for L(n,+14). -Reinhard Zumkeller,Jun 01 2005
Product x*y, where the pair (x, y) solves for x^2 - 3y^2 = -2, i.e., a(n) =A001834(n)*A001835(n). -Lekraj Beedassy,Jul 13 2006
Numbers n such that RootMeanSquare(1,3,...,2*A001570(k)-1) = n. -Ctibor O. Zizka,Sep 04 2008
As n increases, this sequence is approximately geometric with common ratio r = lim(n -> oo, a(n)/a(n-1)) = (2 + sqrt(3))^2 = 7 + 4 * sqrt(3). -Ant King,Nov 15 2011
REFERENCES
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329.
J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.
LINKS
Vincenzo Librandi,Table of n, a(n) for n = 1..890
K. Andersen, L. Carbone, and D. Penta,Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields,Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Alex Fink, Richard K. Guy, and Mark Krusemeyer,Partitions with parts occurring at most thrice,Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
T. N. E. Greville,Table for third-degree spline interpolations with equally spaced arguments,Math. Comp., 24 (1970), 179-183.
W. D. Hoskins,Table for third-degree spline interpolation using equi-spaced knots,Math. Comp., 25 (1971), 797-801.
Tanya Khovanova,Recursive Sequences
E. Kilic, Y. T. Ulutas, and N. Omur,A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters,J. Int. Seq. 14 (2011) #11.5.6, table 4, k=1, t=2.
Dino Lorenzini, and Z. Xiang,Integral points on variable separated curves,Preprint 2016.
F. V. Waugh and M. W. Maxfield,Side-and-diagonal numbers,Math. Mag., 40 (1967), 74-83.
Eric Weisstein's World of Mathematics,Octagonal Square Number.
Index entries for linear recurrences with constant coefficients,signature (14,-1).
FORMULA
a(n) = 2*A001921(n)+1.
a(n) = 14*a(n-1) - a(n-2) for n>1.
a(n) = S(n, 14) + S(n-1, 14) = S(2*n, 4) with S(n, x):= U(n, x/2) Chebyshev's polynomials of the second kind. SeeA049310.S(-1, x) = 0, S(n, 14) =A007655(n+1) and S(n, 4) =A001353(n+1).
G.f.: x*(1+x)/(1-14*x+x^2).
a(n) = (ap^(2*n+1) - am^(2*n+1))/(ap - am) with ap:= 2+sqrt(3) and am:= 2-sqrt(3).
a(n+1) = Sum_{k=0..n} (-1)^k*binomial(2*n-k, k)*16^(n-k), n>=0.
a(n) = sqrt((4*A001570(n-1)^2 - 1)/3).
a(n) ~ 1/6*sqrt(3)*(2 + sqrt(3))^(2*n-1). - Joe Keane (jgk(AT)jgk.org), May 15 2002
4*a(n+1) = (A001834(n))^2 + 4*(A001835(n+1))^2 - (A001835(n))^2. E.g. 4*a(3) = 4*209 = 19^2 + 4*11^2 - 3^2 = (A001834(2))^2 + 4*(A001835(3))^2 - (A001835(2))^2. Generating floretion: 'i + 2'j + 3'k + i' + 2j' + 3k' + 4'ii' + 3'jj' + 4'kk' + 3'ij' + 3'ji' + 'jk' + 'kj' + 4e. -Creighton Dement,Dec 04 2004
a(n) = f(a(n-1),7) + f(a(n-2),7), where f(x,s) = s*x + sqrt((s^2-1)*x^2+1); f(0,s)=0. -Marcos Carreira,Dec 27 2006
FromAnt King,Nov 15 2011: (Start)
a(n) = 1/6 * sqrt(3) * ( (tan(5*Pi/12)) ^ (2n-1) - (tan(Pi/12)) ^ (2n-1) ).
a(n) = floor (1/6 * sqrt(3) * (tan(5*Pi/12)) ^ (2n-1)). (End)
E.g.f.: 1 - exp(7*x)*(3*cosh(4*sqrt(3)*x) - 2*sqrt(3)*sinh(4*sqrt(3)*x))/3. -Stefano Spezia,Dec 12 2022
a(n) = sqrt(A036428(n)). -Bernard Schott,Dec 19 2022
MAPLE
seq(coeff(series((1+x)/(1-14*x+x^2), x, n+1), x, n), n = 0..30); #G. C. Greubel,Dec 06 2019
MATHEMATICA
LinearRecurrence[{14, - 1}, {1, 15}, 17] (*Ant King,Nov 15 2011 *)
CoefficientList[Series[(1+x)/(1-14x+x^2), {x, 0, 30}], x] (*Vincenzo Librandi,Jun 17 2014 *)
PROG
(Sage) [(lucas_number2(n, 14, 1)-lucas_number2(n-1, 14, 1))/12 for n in range(1, 18)] #Zerinvary Lajos,Nov 10 2009
(PARI) Vec((1+x)/(1-14*x+x^2)+O(x^99)) \\Charles R Greathouse IV,Jun 16 2014
(PARI) isok(n) = ispolygonal(n^2, 8); \\Michel Marcus,Jul 09 2017
(Magma) I:=[1, 15]; [n le 2 select I[n] else 14*Self(n-1) - Self(n-2): n in [1..30]]; //G. C. Greubel,Dec 06 2019
(GAP) a:=[1, 15];; for n in [3..30] do a[n]:=14*a[n-1]-a[n-2]; od; a; #G. C. Greubel,Dec 06 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Additional comments fromWolfdieter Lang,Nov 29 2002
Incorrect recurrence relation deleted byAnt King,Nov 15 2011
Minor edits byVaclav Kotesovec,Jan 28 2015
STATUS
approved