OFFSET
0,7
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).
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).
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 *)
PROG
(PARI) a(n) = sum(k=0, n\6, binomial(n-3*k, 3*k)); \\ Michel Marcus, Sep 08 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Nov 06 2004
STATUS
approved