A345457 a(n) = Sum_{k=0..n} binomial(5*n+3,5*k).

1, 57, 1574, 53143, 1669801, 53774932, 1717012749, 54986385093, 1759098789526, 56296324109907, 1801425114687749, 57646238657975068, 1844672594930734801, 59029601136140621857, 1888946370232447241574, 60446293452901248074943

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (21,353,-32).

Formula

G.f.: (1 + 36*x + 24*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).

a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.

a(n) = A139398(5*n+3).

a(n) = 2^(5*n + 4)/10 + (( 475 - 213*sqrt(5))/phi^(5*n) + ( 65 - 33*sqrt(5))*(-1)^n*phi^(5*n)) / (10*(41*sqrt(5)-90)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 20 2021

Mathematica

a[n_] := Sum[Binomial[5*n + 3, 5*k], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Jun 20 2021 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(5*n+3, 5*k));
(PARI) my(N=20, x='x+O('x^N)); Vec((1+36*x+24*x^2)/((1-32*x)*(1+11*x-x^2)))

Crossrefs

Sum_{k=0..n} binomial(b*n+c,b*k): A090408 (b=4,c=3), A070782 (b=5,c=0), A345455 (b=5,c=1), A345456 (b=5,c=2), this sequence (b=5,c=3), A345458 (b=5,c=4).

Cf. A139398.

Sequence in context: A000505 A332911 A179466 * A017773 A017720 A009702

Adjacent sequences: A345454 A345455 A345456 * A345458 A345459 A345460

Keyword

nonn

Author

Seiichi Manyama, Jun 20 2021

Status

approved