A330050 a(n) = 2*((-1)^n - 1)*(F(n) - 1) - (3*(-1)^n + 7)/2 * F(n+1) + 5*F(n+1)^2.

0, 3, 10, 35, 100, 288, 780, 2115, 5610, 14883, 39160, 103040, 270280, 708963, 1857450, 4866435, 12744060, 33373728, 87382900, 228795875, 599019850, 1568318403, 4105974960, 10749749760, 28143378000, 73680759363, 192899171530, 505017737315, 1322154751060

Offset: 0

Author

Michael Somos, Nov 29 2019

Keywords

• nonn
• easy

Examples

			G.f. = 3*x + 10*x^2 + 35*x^3 + 100*x^4 + 288*x^5 + 780*x^6 + 2115*x^7 + ...
		

Links

• V. 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. A203976, A329421, A330051.

Programs

Mathematica

a[ n_] := 2((-1)^n - 1)(Fibonacci[n] - 1) - (3(-1)^n + 7)/2 Fibonacci[n + 1] + 5 Fibonacci[n + 1]^2;
LinearRecurrence[{4,-1,-11,11,1,-4,1},{0,3,10,35,100,288,780},30] (* Harvey P. Dale, Feb 10 2024 *)

PARI

{a(n) = n = abs(n+1)-1; polcoeff( x * O(x^n) + (3*x - 2*x^2 - 2*x^3 + 3*x^4) / (1 - 4*x + x^2 + 11*x^3 - 11*x^4 - x^5 + 4*x^6 - x^7), n)};

Formula

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

a(n) = 2 + L(2*n+2) - F(n+4) - (L(-n+2) + F(n+1))/2 where F=A000045, L=A000032.

a(n) + b(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=A330051.

a(n) = A203976(floor(n/2)+1) * A329421(n).

a(n) = a(-2-n) for all n in Z.