A277287 a(n) = binomial(2*n,n) + Sum_{k=1..n} binomial(2*n-k,n-k)*Fibonacci(k).

1, 3, 10, 36, 133, 499, 1891, 7217, 27690, 106680, 412368, 1598358, 6209542, 24171004, 94246202, 368022472, 1438965885, 5632870627, 22072920103, 86575738469, 339860843589, 1335186464195, 5249164967309, 20650056413491, 81285516680103

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: (2*x+sqrt(1-4*x)+1)/(2*sqrt(1-4*x)*x-8*x+2).

a(n) = A000984(n) + A257838(n).

a(n) ~ 3 * 2^(2*n) / sqrt(Pi*n). - Vaclav Kotesovec, Oct 09 2016

From Vladimir Reshetnikov, Oct 11 2016: (Start)

a(n) = binomial(2*n-1, n-1)*((hypergeom([1, 1-n], [1-2*n], 1-phi)/phi + hypergeom([1, 1-n], [1-2*n], phi)*phi)/sqrt(5) + 2), where phi=(1+sqrt(5))/2.

Recurrence: (n+1)*(n^2-2)*a(n+1) + 2*(2*n^3+n^2-8*n+3)*a(n-2) + (15*n^3+7*n^2-62*n+26)*a(n-1) = 2*(4*n^3+3*n^2-12*n-1)*a(n). (End)

Maple

fib := n -> `if`(n=0, 1, combinat:-fibonacci(n)):
a := n -> add(binomial(2*n-k, n-k)*fib(k), k=0..n):
seq(a(n), n=0..24); # Peter Luschny, Oct 10 2016

Mathematica

Table[Sum[Binomial[2*n-k, n-k]*Fibonacci[k], {k, 1, n}] + Binomial[2*n, n], {n, 0, 20}] (* Vaclav Kotesovec, Oct 09 2016 *)
Round@Table[Binomial[2 n - 1, n - 1] ((Hypergeometric2F1[1, 1 - n, 1 - 2 n, 1 - GoldenRatio]/GoldenRatio + Hypergeometric2F1[1, 1 - n, 1 - 2 n, GoldenRatio] GoldenRatio)/Sqrt[5] + 2), {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Oct 11 2016 *)

Prog

(Maxima) makelist(sum(binomial(2*n-k, n-k)*fib(k), k, 1, n)+binomial(2*n, n), n, 0, 25);
(PARI) a(n) = binomial(2*n, n) + sum(k=1, n, binomial(2*n-k, n-k)*fibonacci(k)); \\ Michel Marcus, Oct 11 2016

Crossrefs

Cf. A000045, A000984, A257838.

Sequence in context: A329533 A102871 A371842 * A119374 A371773 A272686

Adjacent sequences: A277284 A277285 A277286 * A277288 A277289 A277290

Keyword

nonn

Author

Vladimir Kruchinin, Oct 09 2016

Status

approved