A102591 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).

1, 6, 44, 328, 2448, 18272, 136384, 1017984, 7598336, 56714752, 423324672, 3159738368, 23584608256, 176037912576, 1313964867584, 9807567290368, 73204678852608, 546407161659392, 4078438577864704, 30441879976280064

Offset: 0

Author

Paul Barry, Jan 22 2005

Keywords

• easy
• nonn

Comments

In general, Sum_{k=0..n} binomial(2n+1,2k)*r^(n-k) has g.f. (1-(r-1)x)/(1-2(r+1)+(r-1)^2x^2) and a(n) = ((sqrt(r)-1)^(2n+1) + (sqrt(r)+1)^(2n+1))/(2*sqrt(r)).

Links

• Index entries for linear recurrences with constant coefficients, signature (8,-4).

Crossrefs

Bisection of A080879 or A002605.

Cf. A066443, A079935, A099156, A102592, A107903.

Programs

Mathematica

LinearRecurrence[{8,-4},{1,6},20] (* Harvey P. Dale, Sep 28 2021 *)

Formula

G.f.: (1-2x)/(1-8x+4x^2);

a(n) = 8*a(n-1) - 4*a(n-2);

a(n) = sqrt(3)*(sqrt(3)-1)^(2n+1)/6 + sqrt(3)*(sqrt(3)+1)^(2n+1)/6.

a(n) = 2^n*A079935(n). - R. J. Mathar, Sep 20 2012

a(n) = 2^(2*n+1)*Sum_{k >= n} binomial(2*k,2*n)*(1/3)^(k+1). Cf. A099156. - Peter Bala, Nov 29 2021

3*a(n)^2 = A107903(n)^2 + 2^(2*n+1). - Philippe Deléham, Mar 21 2023