A075091 Sum of Lucas numbers and reflected Lucas numbers (comment to A061084).

4, 0, 6, 0, 14, 0, 36, 0, 94, 0, 246, 0, 644, 0, 1686, 0, 4414, 0, 11556, 0, 30254, 0, 79206, 0, 207364, 0, 542886, 0, 1421294, 0, 3720996, 0, 9741694, 0, 25504086, 0, 66770564, 0, 174807606, 0, 457652254, 0, 1198149156, 0, 3136795214, 0, 8212236486, 0, 21499914244, 0

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 31 2002

Keywords

• easy
• nonn
• changed

Comments

a(n) = ((-1)^n + 1)*L(n), where L(n) denotes the n-th Lucas number.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Crossrefs

Cf. A000032, A061084.

Programs

Mathematica

CoefficientList[Series[(4 - 6*x^2)/(1 - 3*x^2 + x^4), {x, 0, 50}], x]
LinearRecurrence[{0,3,0,-1},{4,0,6,0},50] (* or *) Riffle[ LinearRecurrence[ {3,-1},{4,6},30],0] (* Harvey P. Dale, Aug 17 2018 *)

Formula

a(n) = 3*a(n-2) - a(n-4), a(0)=4, a(1)=0, a(2)=6, a(3)=0.

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

E.g.f.: 4*cosh(x/2)*cosh(sqrt(5)*x/2). - Stefano Spezia, Aug 30 2025

Extensions

a(46)-a(49) from Stefano Spezia, Aug 30 2025