A237714 Expansion of (1 + x)/(1 - x^2 - 2*x^5).

1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 11, 13, 21, 23, 35, 45, 61, 87, 107, 157, 197, 279, 371, 493, 685, 887, 1243, 1629, 2229, 2999, 4003, 5485, 7261, 9943, 13259, 17949, 24229, 32471, 44115, 58989, 80013, 107447, 144955, 195677, 262933, 355703, 477827, 645613, 869181, 1171479, 1580587

Offset: 0

Author

Sergio Falcon, Feb 12 2014

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,2).

Crossrefs

Cf. A000129, A122522, A159284.

Programs

Mathematica

For[j = 0, j < 5, j++, a[j] = 1]
For[j = 5, j < 51, j++, a[j] = 2 a[j - 5] + a[j - 2]]
Table[a[j], {j, 0, 50}]
CoefficientList[Series[(1 + x)/(1 - x^2 - 2 x^5), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)
LinearRecurrence[{0,1,0,0,2},{1,1,1,1,1},70] (* Harvey P. Dale, Nov 24 2024 *)

PARI

Vec( (1 + x)/(1 - x^2 - 2*x^5) + O(x^66) ) \\ Joerg Arndt, Feb 24 2014

Formula

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(n) = 2*a(n-5) + a(n-2) for n>=5.

a(2n) = sum_{j=0}^{n/5}C(n-3j,2j)*2^(2j)+sum_{j=0}^{(n-3)/5} C(n-2-3j,2j+1)*2^(2j+1).

a(2n+1) = sum_{j=0}^{n/5}C(n-3j,2j)*2^(2j)+sum_{j=0}^{(n-2)/5} C(n-1-3j,2j+1)*2^(2j+1).