A213046 Convolution of Lucas numbers and positive integers repeated (A000032 and A008619).

2, 3, 8, 13, 25, 41, 71, 116, 193, 314, 514, 834, 1356, 2197, 3562, 5767, 9339, 15115, 24465, 39590, 64067, 103668, 167748, 271428, 439190, 710631, 1149836, 1860481, 3010333, 4870829, 7881179, 12752024, 20633221, 33385262, 54018502, 87403782, 141422304

Offset: 0

Author

Clark Kimberling, Jun 10 2012

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).

Crossrefs

Cf. A213500.

Programs

Magma

/* By definition */ A008619:=func; [&+[A008619(i)*Lucas(n-i): i in [0..n]]: n in [0..34]];

Mathematica

f[x_] := (1 + x) (1 - x)^2; g[x] := 1 - x - x^2;
s = Normal[Series[(2 - x)/(f[x] g[x]), {x, 0, 60}]]
CoefficientList[s, x]  (* A213046 *)
LinearRecurrence[{2,1,-3,0,1},{2,3,8,13,25},40] (* Harvey P. Dale, Aug 31 2023 *)

PARI

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,0,-3,1,2]^n*[2;3;8;13;25])[1,1] \\ Charles R Greathouse IV, Jan 29 2016

PARI

Vec((-2 + x)/((-1 + x)^2*(-1 + 2*x^2 + x^3)) + O(x^60)) \\ Colin Barker, Feb 09 2017

Formula

a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) + a(n-5).

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

a(n) = (-9/4 + (3*(-1)^n)/4 + (2^(-n)*((1-t)^n*(-5+2*t) + (1+t)^n*(5+2*t)))/t + (-1-n)/2) where t=sqrt(5). - Colin Barker, Feb 09 2017