A101890 a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*Fibonacci(k).

0, 0, 1, 3, 7, 15, 32, 70, 157, 357, 815, 1859, 4232, 9620, 21853, 49635, 112747, 256139, 581944, 1322210, 3004145, 6825557, 15507867, 35234183, 80052656, 181881000, 413236953, 938882307, 2133159119, 4846579847, 11011525360

Offset

0,4

Comments

Transform of F(n) under the mapping g(x)-> (1/(1-x))g(x^2/((1-x)^2). Binomial transform of aerated Fibonacci numbers 0,0,1,0,1,0,2,0,3,0,5,...

F(n) may be recovered as Sum_{k=0..2*n} Sum_{j=0..k} C(0,2*n-k)*C(k,j)*(-1)^(k-j)*a(j). - Paul Barry, Jun 10 2005

Links

Table of n, a(n) for n=0..30.

M. Abrate, S. Barbero, U. Cerruti, N. Murru, Fixed Sequences for a Generalization of the Binomial Interpolated Operator and for some Other Operators, J. Int. Seq. 14 (2011) # 11.8.1.

Index entries for linear recurrences with constant coefficients, signature (4,-5,2,1).

Formula

G.f.: x^2*(1-x)/(1-4*x+5*x^2-2*x^3-x^4).

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

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

Mathematica

LinearRecurrence[{4, -5, 2, 1}, {0, 0, 1, 3}, 40] (* Harvey P. Dale, Jul 19 2018 *)

Crossrefs

Cf. A000045.

Sequence in context: A374678 A132402 A137166 * A307573 A134195 A365527

Adjacent sequences: A101887 A101888 A101889 * A101891 A101892 A101893

Keyword

easy,nonn

Author

Paul Barry, Dec 20 2004

Status

approved