A097136 - OEIS

%I #19 May 25 2018 12:56:06

%S 1,4,10,25,64,166,433,1132,2962,7753,20296,53134,139105,364180,953434,

%T 2496121,6534928,17108662,44791057,117264508,307002466,803742889,

%U 2104226200,5508935710,14422580929,37758807076,98853840298,258802713817,677554301152

%N a(n) = 3*Fibonacci(2*n) + 1.

%C Binomial transform of A097135.

%H Colin Barker, <a href="/A097136/b097136.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1)

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

%F a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).

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

%F a(n) = A097132(2*n) = A097133(2*n).

%t Table[3*Fibonacci[2n]+1,{n,0,30}] (* or *) LinearRecurrence[{4,-4,1},{1,4,10},30] (* _Harvey P. Dale_, May 25 2018 *)

%o (PARI) Vec((1-2*x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ _Colin Barker_, Nov 02 2016

%Y Cf. A000045.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Jul 26 2004