OFFSET
0,2
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).
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) + 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) = a(-2-n) for all n in Z.
EXAMPLE
G.f. = 3*x + 10*x^2 + 35*x^3 + 100*x^4 + 288*x^5 + 780*x^6 + 2115*x^7 + ...
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 *)
PROG
(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)};
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Nov 29 2019
STATUS
approved