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a(n) = S(n,3) where S(n,m) = Sum_{k=0..n} binomial(n,k)*fibonacci(m*k).
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%I #27 Jan 01 2024 11:26:47

%S 0,2,12,64,336,1760,9216,48256,252672,1323008,6927360,36272128,

%T 189923328,994451456,5207015424,27264286720,142757658624,747488804864,

%U 3913902194688,20493457948672,107305138913280,561857001684992

%N a(n) = S(n,3) where S(n,m) = Sum_{k=0..n} binomial(n,k)*fibonacci(m*k).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4).

%F a(n) = 6*a(n-1)-4*a(n-2) = 2*A084326(n).

%F a(n) = Sum_{0<=j<=i<=n} C(i,j)*C(n,i)*Fibonacci(i+j). - _Benoit Cloitre_, May 21 2005

%F a(n) = 2^n*Fibonacci(2*n). - _Benoit Cloitre_, Sep 13 2005

%F a(n) = Sum_{k=0..n} C(n,k)*Fibonacci(k)*Lucas(n-k). - _Ross La Haye_, Aug 14 2006

%F G.f.: 2*x/(1-6*x+4*x^2). - _Colin Barker_, Jun 19 2012

%Y Cf. A001906 (S(n, 1)), A030191 (S(n, 2)).

%Y Cf. A084326.

%K nonn,easy

%O 0,2

%A _Benoit Cloitre_, Oct 23 2003