A004799 Self-convolution of Lucas numbers.

1, 6, 17, 38, 80, 158, 303, 566, 1039, 1880, 3364, 5964, 10493, 18342, 31885, 55162, 95032, 163114, 279051, 475990, 809771, 1374316, 2327372, 3933528, 6636025, 11176518, 18794633, 31560206, 52925984, 88646390, 148303719, 247841654

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• É. Czabarka, R. Flórez, and L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
• Index entries for linear recurrences with constant coefficients, signature (2, 1, -2, -1).

Crossrefs

Cf. A000032, A000204, A060922.

Programs

Magma

[((5*n-4)*Lucas(n+1) + 2*Lucas(n))/5: n in [1..30]]; // G. C. Greubel, Dec 17 2017

Maple

a:= n-> (Matrix([[17, 6, 1, 0]]). Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [2, 1, -2, -1][i] else 0 fi)^n) [1,4]: seq (a(n), n=1..40); # Alois P. Heinz, Oct 28 2008

Mathematica

a[n_]:= ((5*n-4)*LucasL[n+1] + 2*LucasL[n])/5; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 12 2015 *)

PARI

Vec(x*((1+2*x)/(1-x-x^2))^2 + O(x^50)) \\ Altug Alkan, Nov 12 2015

Sage

[((5*n-4)*lucas_number2(n+1,1,-1) + 2*lucas_number2(n,1,-1))/5 for n in (1..30)] # G. C. Greubel, Apr 07 2021

Formula

From Wolfdieter Lang, Apr 24 2001: (Start)

a(n) = A060922(n, 1) (second column of Lucas triangle).

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

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

Extensions

More terms from Alois P. Heinz, Oct 28 2008