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

1,2

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).

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)

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 *)

Prog

(PARI) Vec(x*((1+2*x)/(1-x-x^2))^2 + O(x^50)) \\ Altug Alkan, Nov 12 2015
(Magma) [((5*n-4)*Lucas(n+1) + 2*Lucas(n))/5: n in [1..30]]; // G. C. Greubel, Dec 17 2017
(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

Crossrefs

Cf. A000032, A000204, A060922.

Sequence in context: A023621 A000385 A192756 * A085278 A366104 A080275

Adjacent sequences: A004796 A004797 A004798 * A004800 A004801 A004802

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Alois P. Heinz, Oct 28 2008

Status

approved