A100134 - OEIS

%I #33 Jan 05 2025 19:51:37

%S 1,1,1,1,1,1,2,5,11,21,36,57,86,128,194,305,497,827,1381,2287,3739,

%T 6042,9693,15519,24901,40126,64933,105364,171112,277696,450017,728201,

%U 1177181,1902321,3074733,4972113,8044478,13020029,21075947,34114553

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

%C 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

%H Seiichi Manyama, <a href="/A100134/b100134.txt">Table of n, a(n) for n = 0..1000</a>

%H V. C. Harris and C. C. Styles, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/2-4/harris.pdf">A generalization of Fibonacci numbers</a>, Fib. Quart. 2 (1964) 277-289, sequence u(n,3,3).

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,1).

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

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

%p 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

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

%o (PARI) a(n) = sum(k=0, n\6, binomial(n-3*k, 3*k)); \\ _Michel Marcus_, Sep 08 2017

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

%Y Cf. A101551, A102516, A000045.

%K nonn,easy

%O 0,7

%A _Paul Barry_, Nov 06 2004