A053809 Second partial sums of A001891.

1, 6, 21, 57, 133, 281, 554, 1039, 1878, 3302, 5686, 9638, 16143, 26796, 44179, 72471, 118435, 193015, 313920, 509805, 827036, 1340636, 2171996, 3517532, 5695053, 9218786, 14920769, 24147269, 39076593, 63233317, 102320326

Offset: 0

Author

Barry E. Williams, Mar 27 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Crossrefs

Cf. A001911, A001891, A053808.

Right-hand column 9 of triangle A011794. Pairwise sums of A014166.

Programs

GAP

List([0..40], n-> Fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6) # G. C. Greubel, Jul 06 2019

Magma

[Fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6: n in [0..40]]; // G. C. Greubel, Jul 06 2019

Mathematica

Table[Fibonacci[n+10] - (2*n^3+27*n^2+145*n+324)/6, {n,0,40}] (* G. C. Greubel, Jul 06 2019 *)

PARI

vector(40, n, n--; fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6) \\ G. C. Greubel, Jul 06 2019

Sage

[fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6 for n in (0..40)] # G. C. Greubel, Jul 06 2019

Formula

a(n) = a(n-1) + a(n-2) + (2*n+3)*C(n+2, 2)/3; a(-x)=0.

a(n) = Fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6.

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

a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3) + a(n-4) - 3*a(n-5) + a(n-6). - Wesley Ivan Hurt, Apr 21 2021