A058278 Expansion of (1 - x^2)/(1 - x - x^3).

1, 1, 0, 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

Offset: 0

Author

Robert G. Wilson v, Dec 06 2000

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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 A097333.

Cf. A000930.

Programs

Mathematica

CoefficientList[Series[(1 - x^2)/(1 - x - x^3), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Dec 28 2010 *)
LinearRecurrence[{1,0,1},{1,1,0},50] (* Harvey P. Dale, Apr 03 2015 *)

PARI

Vec((1-x^2)/(1-x-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

Formula

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

a(n+3) = Sum_{k=0..n} binomial(n-k, floor(k/2)). - Paul Barry, Jul 06 2004

a(n) = a(n-3) + a(n-1). - Graeme McRae, Apr 26 2010

From Wolfdieter Lang, Apr 21 2015 : (Start)

a(n) = A097333(n-3), n >= 3.

a(n) = A000930(n) - A000930(n-2), n >= 2. (End)

a(n) = A003410(n-4) for n >= 5. - Jianing Song, Aug 11 2023

Extensions

Edited: Offset corrected to 0. The formula by P. Barry corrected. Old formulas adapted to new offset. - Wolfdieter Lang, Apr 21 2015