|
|
A024493
|
|
a(n) = C(n,0) + C(n,3) +... + C(n,3[n/3]).
|
|
34
|
|
|
1, 1, 1, 2, 5, 11, 22, 43, 85, 170, 341, 683, 1366, 2731, 5461, 10922, 21845, 43691, 87382, 174763, 349525, 699050, 1398101, 2796203, 5592406, 11184811, 22369621, 44739242, 89478485, 178956971, 357913942, 715827883, 1431655765, 2863311530
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
a(n) = upper left term of X^n, where X = the 4 X 4 matrix [1,0,1,0; 1,1,0,0; 0,1,1,1; 0,0,0,1]. -Gary W. Adamson,Mar 01 2008
M^n * [1,0,0] = [a(n),A024495(n),A024494(n)], where M = a 3 X 3 matrix [1,1,0; 0,1,1; 1,0,1]. Sum of terms = 2^n. Example: M^5 * [1,0,0] = [11, 11, 10], sum = 2^5 = 32. -Gary W. Adamson,Mar 13 2009
Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1+M)^n = a(n) +A024494(n)*M +A024495(n)*M^2. -Stanislav Sykora,Jun 10 2012
Counts closed walks of length (n) at the vertices of a unidirectional triangle, containing a loop at each vertex. -David Neil McGrath,Sep 15 2014
{A024493,A131708,A024495} is the difference analog of the hyperbolic functions of order 3, {h_1(x), h_2(x), h_3(x)}. For a definition see the reference "Higher Transcendental Functions" and the Shevelev link. -Vladimir Shevelev,Jun 08 2017
|
|
REFERENCES
|
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70.
Higher Transcendental Functions, Bateman Manuscript Project, Vol. 3, ed. A. Erdelyi, 1983 (chapter XVIII).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = (1/3)*(2^n+2*cos( n*Pi/3 )).
G.f.: (1-x)^2/((1-2*x)*(1-x+x^2)) = (1-2*x+x^2)/(1-3*x+3*x^2-2*x^3). -Paul Barry,Feb 11 2004
a(n) = (1/3)*(2^n+b(n)) where b(n) is the 6-periodic sequence {2, 1, -1, -2, -1, 1}. -Benoit Cloitre,May 23 2004
Binomial transform of 1/(1-x^3). G.f.: (1-x)^2/((1-x)^3-x^3) = x/(1-x-2*x^2)+1/(1+x^3); a(n) = Sum_{k=0..floor(n/3)} binomial(n, 3*k); a(n) = Sum_{k=0..n} binomial(n,k)*(cos(2*Pi*k/3+Pi/3)/3+sin(2*Pi*k/3+Pi/3)/sqrt(3)+1/3); a(n) =A001045(n)+sqrt(3)*cos(Pi*n/3+Pi/6)/3+sin(Pi*n/3+Pi*/6)/3+(-1)^n/3. -Paul Barry,Jul 25 2004
a(n) = Sum_{k=0..n} binomial(n, 3*(n-k)). -Paul Barry,Aug 30 2004
G.f.: ((1-x)*(1-x^2)*(1-x^3))/((1-x^6)*(1-2*x)). -Michael Somos,Feb 14 2006
Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) + z(n), y(n+1) = y(n) + x(n), z(n+1) = z(n) + y(n). Then a(n) = x(n). -Stanislav Sykora,Jun 10 2012
E.g.f.: (exp(2*z)+2*cos(z*sqrt(3/4))*exp(z/2))/3. -Peter Luschny,Jul 10 2012
|
|
MAPLE
|
A024493_list:= proc(n) local i; series((exp(2*z)+2*cos(z*sqrt(3/4))*exp(z/2)) /3, z, n+2): seq(i!*coeff(%, z, i), i=0..n) end:A024493_list(33); #Peter Luschny,Jul 10 2012
seq((3*(-1)^(floor((n+1)/3))+(-1)^n+2^(n+1))/6, n=0..33); #Peter Luschny,Jun 14 2017
|
|
MATHEMATICA
|
nn = 18; a = Sum[x^(3 i)/(3 i)!, {i, 0, nn}]; b = Exp[x]; Range[0, nn]! CoefficientList[Series[a b, {x, 0, nn}], x] (*Geoffrey Critzer,Dec 27 2011 *)
Differences[LinearRecurrence[{3, -3, 2}, {0, 1, 2}, 40]] (*Harvey P. Dale,Nov 27 2013 *)
|
|
PROG
|
(PARI) a(n)=sum(i=0, n, sum(j=0, n, if(n-i-3*j, 0, n!/(i)!/(3*j)!)))
(PARI) a(n)=sum(k=0, n\3, binomial(n, 3*k)) /*Michael Somos,Feb 14 2006 */
(PARI) a(n)=if(n<0, 0, ([1, 0, 1; 1, 1, 0; 0, 1, 1]^n)[1, 1]) /*Michael Somos,Feb 14 2006 */
(Magma) I:=[1, 1, 1]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+2*Self(n-3): n in [1..50]]; //Vincenzo Librandi,Jun 12 2017
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|