%I #20 Sep 08 2022 08:46:16

%S 5,16,43,113,296,775,2029,5312,13907,36409,95320,249551,653333,

%T 1710448,4478011,11723585,30692744,80354647,210371197,550758944,

%U 1441905635,3774957961,9882968248,25873946783,67738872101,177342669520,464289136459,1215524739857

%N a(n) = k*Fibonacci(2*n+1) + (k+1)*Fibonacci(2*n), where k=5.

%H Colin Barker, <a href="/A271359/b271359.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).

%F G.f.: (5+x) / (1-3*x+x^2).

%F a(n) = 3*a(n-1)-a(n-2) for n>1.

%F a(n) = (2^(-2-n)*((13-sqrt(5))*(3+sqrt(5))^(n+1) - (13+sqrt(5))*(3-sqrt(5))^(n+1))) / sqrt(5).

%F a(n) = 6*Fibonacci(2*n+2) - Fibonacci(2*n+1) = 5*A001906(n+1) +A001906(n).

%o (PARI) a(n) = 5*fibonacci(2*n+1) + 6*fibonacci(2*n)

%o (PARI) Vec((5+x)/(1-3*x+x^2) + O(x^50))

%o (Magma) k:=5; [k*Fibonacci(2*n+1)+(k+1)*Fibonacci(2*n): n in [0..30]]; // _Bruno Berselli_, Apr 06 2016

%Y Cf. A000045.

%Y Cf. A001906 (k=0), A002878 (k=1), A100545 (k=2, without the initial 2), A271357 (k=3), A271358 (k=4), this sequence (k=5).

%K nonn,easy

%O 0,1

%A _Colin Barker_, Apr 05 2016

%E Changed offset and adapted definition, programs and formulas by _Bruno Berselli_, Apr 06 2016