A330050 - OEIS

%I #18 Feb 10 2024 17:47:40

%S 0,3,10,35,100,288,780,2115,5610,14883,39160,103040,270280,708963,

%T 1857450,4866435,12744060,33373728,87382900,228795875,599019850,

%U 1568318403,4105974960,10749749760,28143378000,73680759363,192899171530,505017737315,1322154751060

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

%H V. Kotesovec, <a href="https://mathematica.stackexchange.com/q/206655">Why is this product equal to zero, when the correct result is 2+GoldenRatio</a>, Mathematica StackExchange, Sep 22 2019.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-11,11,1,-4,1).

%F 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).

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

%F 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.

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

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

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

%t a[ n_] := 2((-1)^n - 1)(Fibonacci[n] - 1) - (3(-1)^n + 7)/2 Fibonacci[n + 1] + 5 Fibonacci[n + 1]^2;

%t LinearRecurrence[{4,-1,-11,11,1,-4,1},{0,3,10,35,100,288,780},30] (* _Harvey P. Dale_, Feb 10 2024 *)

%o (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)};

%Y Cf. A203976, A329421, A330051.

%K nonn,easy

%O 0,2

%A _Michael Somos_, Nov 29 2019