A074392 a(n) = Lucas(n+1) + (3*(-1)^n - 1)/2.

2, 1, 5, 5, 12, 16, 30, 45, 77, 121, 200, 320, 522, 841, 1365, 2205, 3572, 5776, 9350, 15125, 24477, 39601, 64080, 103680, 167762, 271441, 439205, 710645, 1149852, 1860496, 3010350, 4870845, 7881197, 12752041, 20633240, 33385280, 54018522

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 22 2002

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A000032, A074331.

Cf. A004146.

Programs

Mathematica

CoefficientList[Series[(2-x)/(1-x-2*x^2+x^3+x^4), {x, 0, 40}], x]

PARI

Vec((2-x) / ((x-1)*(1+x)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, Jul 12 2017

Formula

a(n) = Sum (L(2i+e), (i=0, 1, .., Floor(n/2))), where L(n) are Lucas numbers and e=2(n/2 - Floor(n/2)).

Convolution of L(n) with the sequence (1, 0, 1, 0, 1, 0, ...)

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>3.

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

a(n) = 2*A052952(n)-A052952(n-1). - R. J. Mathar, Oct 04 2013

a(n) = 2^(-1-n) * (3*(-2)^n - 2^n + (1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n)). - Colin Barker, Jul 12 2017