A247526 a(n) = L(n+1) * L(n) * L(n-1) * L(n-2) / 6, where L(n) = Lucas numbers (A000032).

-1, -1, 4, 14, 154, 924, 6699, 44979, 310764, 2123554, 14571974, 99833524, 684385079, 4690541639, 32150245204, 220358978774, 1510368355474, 10352204457804, 70955102255139, 486333408161979, 3333379024971324, 22847319059525674, 156597856242950654

Offset: 0

Author

Michael Somos, Sep 19 2014

Keywords

• sign
• easy

Examples

			G.f. = -1 - x + 4*x^2 + 14*x^3 + 154*x^4 + 924*x^5 + 6699*x^6 + 44979*x^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Crossrefs

Cf. A000032, A085696.

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-(1-4*x-24*x^2+6*x^3-4*x^4)/((1-x)*(1+3*x+x^2)*(1 - 7*x+x^2)))); // G. C. Greubel, Aug 05 2018

Mathematica

CoefficientList[Series[-(1-4*x-24*x^2+6*x^3-4*x^4)/((1-x)*(1+3*x+x^2)*(1 - 7*x+x^2)), {x, 0, 60}], x] (* G. C. Greubel, Aug 05 2018 *)
Times@@#/6&/@Partition[LucasL[Range[-2,30]],4,1] (* or *) LinearRecurrence[{5,15,-15,-5,1},{-1,-1,4,14,154},30] (* Harvey P. Dale, Apr 20 2022 *)

PARI

{a(n) = my(u = fibonacci(n), v = fibonacci(n-1)); (3*u + v) * (u + 2*v) * (2*u - v) * (-u + 3*v) / 6};

PARI

{a(n) = if( n<1, n=1-n); polcoeff( - (1 - 4*x - 24*x^2 + 6*x^3 - 4*x^4) / ((1 - x) * (1 + 3*x + x^2) * (1 - 7*x +x^2)) + x * O(x^n), n)};

Formula

G.f.: -(1 - 4*x - 24*x^2 + 6*x^3 - 4*x^4)/((1-x)*(1 + 3*x + x^2)*(1 - 7*x + x^2)).

a(n) = a(1-n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5) for all n in Z.

0 = a(n)*(+a(n+1) - 2*a(n+2)) + a(n+1)*(-5*a(n+1) + a(n+2)) for all n in Z.

From Klaus Purath, Oct 02 2020: (Start)

a(n) = (L(n-2)*L(n-1)^3 - L(n-1)*L(n-2)^3)/6 where L = Lucas.

a(n) = f(n-3)*f(n-2)*f(n-1)*f(n)/96 where f = A022112.

a(n) = (f(n-2)*f(n-1)^3 - f(n-1)*f(n-2)^3)/96 where f = A022112.

(End)

2*a(n) = A098149(n) +A004187(n+1)-6*A004187(n) -2 . - R. J. Mathar, Sep 24 2021