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A082988 a(n) = Sum_{k=0..n} 4^k*F(k) where F(k) is the k-th Fibonacci number.

%I A082988 #27 Apr 29 2025 04:43:25
%S A082988 0,4,20,148,916,6036,38804,251796,1628052,10540948,68212628,441505684,
%T A082988 2857424788,18493790100,119693957012,774676469652,5013809190804,
%U A082988 32450060277652,210021188163476,1359285717096340,8797481879000980
%N A082988 a(n) = Sum_{k=0..n} 4^k*F(k) where F(k) is the k-th Fibonacci number.
%C A082988 More generally for any complex number z, the 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 A082988 Seiichi Manyama, <a href="/A082988/b082988.txt">Table of n, a(n) for n = 0..1000</a>
%H A082988 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,12,-16).
%F A082988 a(0) = 0, a(1) = 4, a(2) = 20, a(n) = 5a(n-1)+12a(n-2)-16a(n-3).
%F A082988 O.g.f.: 4*x/((x-1)*(16*x^2+4*x-1)). - _R. J. Mathar_, Dec 05 2007
%t A082988 LinearRecurrence[{5, 12, -16}, {0, 4, 20}, 21] (* _Amiram Eldar_, Apr 29 2025 *)
%o A082988 (PARI) a(n)=if(n<0,0,sum(k=0,n,fibonacci(k)*4^k));
%Y A082988 Cf. A000045, A014334, A082987, A119282.
%K A082988 nonn,easy
%O A082988 0,2
%A A082988 _Benoit Cloitre_, May 29 2003