A100134 a(n) = Sum_{k=0..floor(n/6)} binomial(n-3k,3k).

1, 1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 57, 86, 128, 194, 305, 497, 827, 1381, 2287, 3739, 6042, 9693, 15519, 24901, 40126, 64933, 105364, 171112, 277696, 450017, 728201, 1177181, 1902321, 3074733, 4972113, 8044478, 13020029, 21075947, 34114553

Offset: 0

Author

Paul Barry, Nov 06 2004

Keywords

• nonn
• easy

Comments

For n > 1, a(n-1) + A101551(n-1) + A102516(n-2) = F(n) where F(n) is the n-th Fibonacci number (A000045(n)). This sequence, A101551 and A102516 can be viewed as parts of a three-term linear recurrence defined as b(0) = b(1) = (1,0,0) = (x(0),y(0),z(0)) = (x(1),y(1),z(1)); b(n+1) = (x(n)+y(n-1),y(n)+z(n-1),z(n)+x(n-1)); which gives a(n) = x(n), A101551(n) = y(n), A102516(n) = z(n+1). - Gerald McGarvey, Apr 26 2005

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,3,3).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,1).

Crossrefs

Cf. A100135, A100136, A100137, A100138, A100139.

Cf. A101551, A102516, A000045.

Programs

Maple

ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,X,X))), X = Sequence(b,card >= 2)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=2..41); # Zerinvary Lajos, Mar 26 2008

Mathematica

Table[Sum[Binomial[n-3k,3k],{k,0,Floor[n/6]}],{n,0,40}] (* Harvey P. Dale, Sep 22 2020 *)

PARI

a(n) = sum(k=0, n\6, binomial(n-3*k, 3*k)); \\ Michel Marcus, Sep 08 2017

Formula

G.f.: (1-x)^2/((1-x)^3 - x^6);

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