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%I #39 Jan 05 2025 19:51:41
%S 0,2,8,48,240,1280,6656,34944,182784,957440,5012480,26247168,
%T 137428992,719593472,3767828480,19728629760,103300399104,540888006656,
%U 2832126181376,14829205585920,77646727741440,406563546202112,2128794362052608,11146511995895808
%N a(n) = 2^n*F(n)*F(n+1), where F = A000045.
%H G. C. Greubel, <a href="/A292277/b292277.txt">Table of n, a(n) for n = 0..1000</a>
%H R. S. Melham, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/55-2/MelhamWeightedSums11272016.pdf">Closed Forms for Finite Sums of Weighted Products of Generalized Fibonacci Numbers</a>, The Fibonacci Quarterly 55 (May 2017), Number 2, pages 99-104.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,8,-8).
%F G.f.: 2*x/((1 + 2*x)*(1 - 6*x + 4*x^2)).
%F a(n) = 4*a(n-1) + 8*a(n-2) - 8*a(n-3).
%F a(n) = ((1+sqrt(5))^(2*n+1) + (1-sqrt(5))^(2*n+1))/(10*2^n) - (-2)^n/5, therefore 5*a(n) + (-2)^n = A082762(n). - _Bruno Berselli_, Sep 13 2017
%t Table[2^n Fibonacci[n] Fibonacci[n+1], {n, 0, 40}]
%t Table[((1 + Sqrt[5])^(2 n + 1) + (1 - Sqrt[5])^(2 n + 1))/(10 2^n) - (-2)^n/5, {n, 0, 30}] (* _Bruno Berselli_, Sep 13 2017 *)
%o (Magma) [2^n*Fibonacci(n)*Fibonacci(n+1): n in [0..30]];
%o (PARI) a(n) = 2^n*fibonacci(n)*fibonacci(n+1); \\ _Altug Alkan_, Sep 13 2017
%o (Sage) [2^n*fibonacci(n)*fibonacci(n+1) for n in range(30)] # _Bruno Berselli_, Sep 13 2017
%Y Cf. A000045, A000079, A001654, A082762.
%K nonn,easy
%O 0,2
%A _Vincenzo Librandi_, Sep 13 2017