A097132 a(n) = Sum_{k=0..n} Fibonacci(k) + (-1)^k*Fibonacci(k-1).

1, 2, 4, 5, 10, 12, 25, 30, 64, 77, 166, 200, 433, 522, 1132, 1365, 2962, 3572, 7753, 9350, 20296, 24477, 53134, 64080, 139105, 167762, 364180, 439205, 953434, 1149852, 2496121, 3010350, 6534928, 7881197, 17108662, 20633240, 44791057

Offset: 0

Author

Paul Barry, Jul 26 2004

Keywords

• easy
• nonn

Comments

Partial sums of A097131.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-1,1).

Crossrefs

Cf. A000032, A000045, A097131, A097136.

Programs

Mathematica

LinearRecurrence[{1,3,-3,-1,1},{1,2,4,5,10},40] (* Harvey P. Dale, Nov 12 2022 *)

Formula

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

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

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);

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

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