A330051 a(n) = 1 + F(2*n+1) - (F(n+4) - (-1)^n*F(n-2))/2 where F=A000045.

0, 0, 2, 7, 25, 72, 208, 564, 1530, 4059, 10769, 28336, 74560, 195576, 513010, 1344063, 3521385, 9221688, 24149456, 63230860, 165558250, 433454835, 1134845857, 2971111392, 7778592000, 20364739632, 53315898338, 139583151799, 365434267705, 956720165544

Offset: 0

Author

Michael Somos, Dec 01 2019

Keywords

• nonn
• easy

Examples

			G.f. = 2*x^2 + 7*x^3 + 25*x^4 + 72*x^5 + 208*x^6 + 564*x^7 + 1530*x^8 + ...
		

Links

• Vaclav Kotesovec, Why is this product equal to zero, when the correct result is 2+GoldenRatio, Mathematica StackExchange, Sep 22 2019.
• Index entries for linear recurrences with constant coefficients, signature (4,-1,-11,11,1,-4,1).

Crossrefs

Cf. A000045, A005013, A329421, A330050.

Programs

Mathematica

a[n_] := 1 + Fibonacci[2 n + 1] - (Fibonacci[n + 4] - (-1)^n Fibonacci[n - 2])/2

PARI

{a(n) = 1 + fibonacci(2*n + 1) - (fibonacci(n + 4) - (-1)^n*fibonacci(n - 2))/2};

Formula

a(n) = 1 + F(2*n+1) - F(n+2) - (F(-n+2) + F(n+1))/2.

G.f.: (2*x^2 - x^3 - x^4 + x^5) / (1 - 4*x + x^2 + 11*x^3 - 11*x^4 - x^5 + 4*x^6 - x^7).

b(n) + a(n) * sqrt(5) = F(2*n+2) * Product_{k=2..n} 1 / (1 - q^k/(1 - q^(2*k))) where q = (sqrt(5)-1)/2 and b=A330050.

a(n) = A005013(floor(n/2)) * A329421(n).

Extensions

Definition corrected by N. J. A. Sloane, May 29 2022 following a suggestion from Kevin Ryde.

Additional corrections by Eric Rowland, May 31 2022