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a(n) = (Fibonacci(2*n)-(-1)^n*Fibonacci(n))/2.
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%I #27 Jul 07 2017 13:59:06

%S 0,1,1,5,9,30,68,195,483,1309,3355,8900,23112,60813,158717,416325,

%T 1088661,2852242,7463884,19546175,51163695,133962621,350695511,

%U 918170280,2403740304,6293172025,16475579353,43133883845,112925557953

%N a(n) = (Fibonacci(2*n)-(-1)^n*Fibonacci(n))/2.

%C A049602 gives the coefficients of x in the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1. For the constant terms, see A192352. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232. - _Clark Kimberling_, Jun 29 2011

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-4,1).

%F a(n)=Sum{T(2i+1, n-2i-1): i=0, 1, ..., [ (n+1)/2 ]}, array T as in A049600.

%F Cosh transform of Fibonacci numbers A000045 (or mean of binomial and inverse binomial transforms of A000045). E.g.f.: cosh(x)(2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2). - _Paul Barry_, May 10 2003

%F a(n)=sum{k=0..floor(n/2), C(n, 2k)Fib(n-2k)}; - _Paul Barry_, May 01 2005

%F a(n)=2a(n-1)+3a(n-2)-4a(n-3)+a(n-4). - _Paul Curtz_, Jun 16 2008

%F G.f.: x(1-x)/((1+x-x^2)(1-3x+x^2)); a(n)=sum{k=0..n-1, (-1)^(n-k+1)*F(2k+2)*F(n-k+1)}; - _Paul Barry_, Jul 11 2008

%t LinearRecurrence[{2,3,-4,1},{0,1,1,5},30] (* _Harvey P. Dale_, Jul 07 2017 *)

%o (PARI) a(n)=(fibonacci(2*n)-(-1)^n*fibonacci(n))/2 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A049601.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_

%E Simpler description from _Vladeta Jovovic_ and _Thomas Baruchel_, Aug 24 2004

%E More terms from _Paul Curtz_, Jun 16 2008