A113032 a(n) = Sum_{k=0..floor(n/8)} binomial(n-5*k, 3*k).

1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 57, 85, 121, 167, 228, 315, 449, 666, 1023, 1605, 2533, 3974, 6156, 9394, 14137, 21051, 31159, 46066, 68305, 101850, 152857, 230720, 349576, 530476, 804579, 1217951, 1838897, 2769267, 4161918, 6247570, 9375799

Offset

0,9

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,1).

Formula

G.f.: (1-x)^2/(1-3*x+3*x^2-x^3-x^8). [corrected by Georg Fischer, Apr 17 2020]

Example

a(10+1) = 11 because C(10,0) + C(5,3) = 1+10 = 11.

Mathematica

Table[Sum[Binomial[n - 5*k, 3*k], {k, 0, Floor[n/8]}], {n, 0, 50}] (* G. C. Greubel, Apr 09 2018 *)

Prog

(PARI) a(n) = sum(k=0, n\8, binomial(n-5*k, 3*k)); \\ Michel Marcus, Sep 05 2013
(PARI) lista(nn) = {my(x = xx + O(xx^nn)); gf = (1-x)^2/(1-3*x+3*x^2-x^3-x^8); for (i=0, nn-1, print1(polcoeff(gf, i, xx), ", ")); } \\ Michel Marcus, Sep 05 2013
(Magma) [(&+[Binomial(n-5*k, 3*k): k in [0..Floor(n/8)]]): n in [0..50]]; // G. C. Greubel, Apr 09 2018
(SageMath) ((1-x)^2/(1-3*x+3*x^2-x^3-x^8)).series(x, 44).coefficients(x, sparse=False) # Stefano Spezia, Aug 19 2023

Crossrefs

Cf. A005251, A005253, A000045.

Cf. A050407 (differs from a(16)=167 on), A000292 (essentially the same as A050407-1).

Sequence in context: A352234 A332063 A050407 * A370722 A100134 A365732

Adjacent sequences: A113029 A113030 A113031 * A113033 A113034 A113035

Keyword

nonn,easy

Author

Alexey Kistanov (plast(AT)solid.ru), Jan 05 2006

Extensions

Corrected by T. D. Noe, Nov 01 2006

More terms from Michel Marcus, Sep 05 2013

Status

approved