A101890 - OEIS

%I #17 Aug 31 2024 17:39:00

%S 0,0,1,3,7,15,32,70,157,357,815,1859,4232,9620,21853,49635,112747,

%T 256139,581944,1322210,3004145,6825557,15507867,35234183,80052656,

%U 181881000,413236953,938882307,2133159119,4846579847,11011525360

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

%C 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,...

%C 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

%H M. Abrate, S. Barbero, U. Cerruti, N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barbero/barbero9.html"> Fixed Sequences for a Generalization of the Binomial Interpolated Operator and for some Other Operators</a>, J. Int. Seq. 14 (2011) # 11.8.1.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2,1).

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

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

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

%t LinearRecurrence[{4,-5,2,1},{0,0,1,3},40] (* _Harvey P. Dale_, Jul 19 2018 *)

%Y Cf. A000045.

%K easy,nonn

%O 0,4

%A _Paul Barry_, Dec 20 2004