A060929 Second convolution of Lucas numbers A000032(n+1), n >= 0.

1, 9, 39, 120, 315, 753, 1687, 3612, 7470, 15040, 29634, 57366, 109421, 206115, 384105, 709152, 1298613, 2360943, 4264835, 7659870, 13686456, 24340184, 43102644, 76031100, 133636825, 234116493, 408900987

Offset: 0

Author

Wolfdieter Lang, Apr 20 2001

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Crossrefs

Cf. A004799, A060922.

Programs

Magma

I:=[1,9,39,120,315,753]; [n le 6 select I[n] else 3*Self(n-1) - 5*Self(n-3) + 3*Self(n-5) + Self(n-6): n in [1..30]]; // G. C. Greubel, Dec 21 2017

Mathematica

CoefficientList[Series[((1 + 2*x)/(1 - x - x^2))^3, {x, 0, 50}], x] (* or *) LinearRecurrence[{3,0,-5,0,3,1}, {1,9,39,120,315,753}, 30] (* G. C. Greubel, Dec 21 2017 *)

PARI

x='x+O('x^30); Vec(((1+2*x)/(1-x-x^2))^3) \\ G. C. Greubel, Dec 21 2017

Formula

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

a(n) = A060922(n+2, 2) (third column of Lucas triangle).

a(n) = (n+1)*((5*n+4)*L(n+2) + (5*n-2)*L(n+1))/10, n >= 1, with the Lucas numbers L(n)=A000032(n)=A000204(n), n >= 1.