A082988 - OEIS

A082988

a(n) = Sum_{k=0..n} 4^k*F(k) where F(k) is the k-th Fibonacci number.

%I #19 Oct 03 2023 13:11:25

%S 0,4,20,148,916,6036,38804,251796,1628052,10540948,68212628,441505684,

%T 2857424788,18493790100,119693957012,774676469652,5013809190804,

%U 32450060277652,210021188163476,1359285717096340,8797481879000980

%N a(n) = Sum_{k=0..n} 4^k*F(k) where F(k) is the k-th Fibonacci number.

%C More generally for any complex number z, sequence a(n)=Sum_{k=0..n} z^k*F(k) satisfies the recurrence : a(0)=0, a(1)=z, a(2)=z(z+1), for n>2 a(n)=(z+1)*a(n-1)+z*(z-1)*a(n-2)-z^2*a(n-3)

%H Seiichi Manyama, <a href="/A082988/b082988.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,12,-16).

%F a(0)=0, a(1)=4, a(2)=20, a(n)=5a(n-1)+12a(n-2)-16a(n-3).

%F O.g.f.: 4*x/((x-1)*(16*x^2+4*x-1)). - _R. J. Mathar_, Dec 05 2007

%o (PARI) a(n)=if(n<0,0,sum(k=0,n,fibonacci(k)*4^k))

%Y Cf. A014334, A082987, A119282.

%K nonn,easy

%O 0,2

%A _Benoit Cloitre_, May 29 2003