A301696 Partial sums of A219529.

1, 6, 17, 33, 54, 81, 113, 150, 193, 241, 294, 353, 417, 486, 561, 641, 726, 817, 913, 1014, 1121, 1233, 1350, 1473, 1601, 1734, 1873, 2017, 2166, 2321, 2481, 2646, 2817, 2993, 3174, 3361, 3553, 3750, 3953, 4161, 4374, 4593, 4817, 5046, 5281, 5521, 5766

Offset: 0

Author

N. J. A. Sloane, Mar 25 2018

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A219529.

Programs

Maple

A301696:= n-> (8*(3*n*(n+1) +1) + `mod`(n+2, 3) - `mod`(n+1, 3))/9;
seq(A301696(n), n=0..60); # G. C. Greubel, May 27 2020

Mathematica

Table[(Mod[n+2, 3] - Mod[n+1, 3] + 8*(3*n*(n+1) +1))/9, {n,0,60}] (* G. C. Greubel, May 27 2020 *)

PARI

Vec((1 + x)^4 / ((1 - x)^3*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, Mar 26 2018

Sage

[(24*n*(n+1)+8 + (n+2)%3 - (n+1)%3 )/9 for n in (0..60)] # G. C. Greubel, May 27 2020

Formula

From Colin Barker, Mar 26 2018: (Start)

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

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

From G. C. Greubel, May 27 2020: (Start)

a(n) = (ChebyshevU(n, -1/2) - ChebyshevU(n-1, -1/2) + 8*(3*n*(n+1) +1))/9.

a(n) = ( A131713(n) + 8*A028896(n) + 8 )/9. (End)