A097132 - OEIS

%I #14 Nov 12 2022 13:48:37

%S 1,2,4,5,10,12,25,30,64,77,166,200,433,522,1132,1365,2962,3572,7753,

%T 9350,20296,24477,53134,64080,139105,167762,364180,439205,953434,

%U 1149852,2496121,3010350,6534928,7881197,17108662,20633240,44791057

%N a(n) = Sum_{k=0..n} Fibonacci(k) + (-1)^k*Fibonacci(k-1).

%C Partial sums of A097131.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-1,1).

%F G.f.: (1 + x - x^2 - 2*x^3)/((1 - 3*x^2 + x^4)*(1-x));

%F a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - a(n-4) + a(n-5);

%F a(n) = 1 + (1/2 - sqrt(5)/2)^n*(1/2 - 3*sqrt(5)/10) - (sqrt(5)/2 - 1/2)^n*(3*sqrt(5)/10 + 1/2) + (-sqrt(5)/2 - 1/2)^n*(3*sqrt(5)/10 - 1/2) + (sqrt(5)/2 + 1/2)^n*(3*sqrt(5)/10 + 1/2);

%F a(2n) = 1 + 3*Fibonacci(2n) = A097136(n);

%F a(2n+1) = 1 + Fibonacci(2n) + Fibonacci(2n+2) = 1 + Lucas(2n).

%t LinearRecurrence[{1,3,-3,-1,1},{1,2,4,5,10},40] (* _Harvey P. Dale_, Nov 12 2022 *)

%Y Cf. A000032, A000045, A097131, A097136.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 26 2004