A097333 a(n) = Sum_{k=0..n} C(n-k, floor(k/2)).

1, 2, 2, 3, 5, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982, 1427440, 2092015, 3065997, 4493437, 6585452

Offset: 0

Author

Paul Barry, Aug 05 2004

Keywords

• easy
• nonn

Comments

Partial sums of A058278.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Kassie Archer and Aaron Geary, Powers of permutations that avoid chains of patterns, arXiv:2312.14351 [math.CO], 2023. See p. 15.
• Engin Özkan, Bahar Kuloǧlu, and James Peters, k-Narayana sequence self-similarity, hal-03242990 [math.CO], 2021. See p. 12.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Crossrefs

Essentially the same as A003410 and A058278.

Programs

Mathematica

LinearRecurrence[{1, 0, 1}, {1, 2, 2}, 70] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011*)

PARI

a(n) = sum(k=0, n, binomial(n-k, k\2)); \\ Michel Marcus, Mar 02 2022

Formula

G.f.: (1+x-x^2-x^3)/((1-x)*(1-x^2-x^3-x^4)) = (1+x)/(1-x-x^3);

a(n) = a(n-1) + a(n-3);

a(n) = a(n-1) + a(n-2) - a(n-5).

a(n) = A058278(n+3) = A000930(n-1)+A000930(n). - R. J. Mathar, Jul 07 2023

a(n) = A003410(n-1) for n >= 2. - Jianing Song, Aug 11 2023