A301695 Expansion of (1 + 5*x + 4*x^2 + 5*x^3 + x^4)/((1 - x)^2*(1 - x^3)).

1, 7, 17, 33, 55, 81, 113, 151, 193, 241, 295, 353, 417, 487, 561, 641, 727, 817, 913, 1015, 1121, 1233, 1351, 1473, 1601, 1735, 1873, 2017, 2167, 2321, 2481, 2647, 2817, 2993, 3175, 3361, 3553, 3751, 3953, 4161, 4375, 4593, 4817, 5047, 5281, 5521, 5767, 6017, 6273

Offset: 0

Author

N. J. A. Sloane, Mar 25 2018

Keywords

• nonn
• easy

Links

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

Crossrefs

Partial sums of A301694.

Programs

Magma

[(8*n*(n+1)-2*((n-1)^2 mod 3)+5)/3: n in [0..50]]; // Bruno Berselli, Mar 26 2018

Mathematica

Table[(8 n (n + 1) -  2 ((n-1)^2 mod 3) + 5)/3, {n, 0, 40}] (* Bruno Berselli, Mar 26 2018 *)

PARI

Vec((1 + 5*x + 4*x^2 + 5*x^3 + x^4)/((1 - x)^2*(1 - x^3)) + O(x^50)) \\ Felix Fröhlich, Mar 26 2018

Formula

G.f.: (1 + 5*x + 4*x^2 + 5*x^3 + x^4)/((1 - x)^2*(1 - x^3)).

a(n) = (8*n*(n + 1) - 2*((n - 1)^2 mod 3) + 5)/3. Therefore: a(3*k + r) = 8*k*(3*k + 2*r + 1) + 8*r + (-1)^r. Example: a(13) = a(3*4+1) = 8*4*(3*4 + 2*1 + 1) + 8*1 + (-1)^1 = 487. - Bruno Berselli, Mar 26 2018