A075150 a(n) = L(n)*C(n), L(n)=Lucas numbers (A000032), C(n)=reflected Lucas numbers (see comment to A061084).

4, -1, 9, -16, 49, -121, 324, -841, 2209, -5776, 15129, -39601, 103684, -271441, 710649, -1860496, 4870849, -12752041, 33385284, -87403801, 228826129, -599074576, 1568397609, -4106118241, 10749957124, -28143753121, 73681302249, -192900153616, 505019158609, -1322157322201

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Sep 05 2002

Keywords

• easy
• sign

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,2,1).

Crossrefs

Cf. A000032, A001254, A061084, A219233.

Programs

Magma

A075150:= func< n | (-1)^n*Lucas(n)^2 >; // G. C. Greubel, Jun 14 2025

Mathematica

CoefficientList[Series[(4 + 7*x - x^2)/(1 + 2*x - 2*x^2 - x^3), {x, 0, 30}], x]
LinearRecurrence[{-2,2,1},{4,-1,9},50] (* Harvey P. Dale, Nov 08 2011 *)

PARI

a(n) = round((2+(1/2*(-3-sqrt(5)))^n+(1/2*(-3+sqrt(5)))^n)) \\ Colin Barker, Oct 01 2016

PARI

Vec((4+7*x-x^2)/(1+2*x-2*x^2-x^3) + O(x^30)) \\ Colin Barker, Oct 01 2016

SageMath

def A075150(n): return (-1)**n*lucas_number2(n,1,-1)**2 # G. C. Greubel, Jun 14 2025

Formula

a(n) = (-1)^n*A000032(2*n) + 2.

a(n) = -2*a(n-1) + 2*a(n-2) + a(n-3) with a(0)=4, a(1)=-1, a(2)=9.

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

a(n) = (-1)^n*A001254(n). - R. J. Mathar, Jan 11 2012

a(n) = 2 + (1/2*(-3-sqrt(5)))^n + (1/2*(-3+sqrt(5)))^n. - Colin Barker, Oct 01 2016

From G. C. Greubel, Jun 14 2025: (Start)

a(n) = A000032(n)*A000032(-n) = (-1)^n*A000032(n)^2.

a(n) = A219233(n) + 2 + [n=0].

E.g.f.: 2*exp(-3*x/2)*cosh(sqrt(5)*x/2) + 2*exp(x). (End)