A099924 Self-convolution of Lucas numbers.

4, 4, 13, 22, 45, 82, 152, 274, 491, 870, 1531, 2676, 4652, 8048, 13865, 23798, 40713, 69446, 118144, 200510, 339559, 573894, 968183, 1630632, 2742100, 4604572, 7721797, 12933334, 21637221, 36159610, 60367976, 100687786

Offset

0,1

References

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 57.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..4767

É. Czabarka, R. Flórez, and L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.

Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.

Tamás Szakács, Convolution of second order linear recursive sequences. II. Commun. Math. 25, No. 2, 137-148 (2017). See remark 4.

Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 37.

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Formula

a(n) = (n+1)*L(n) + 2F(n+1) = Sum_{k=0..n} L(k)*L(n-k).

G.f.: (2-x)^2/(1-x-x^2)^2, corrected Aug 23 2022

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4), a(0)=4, a(1)=4, a(2)=13, a(3)=22. - Harvey P. Dale, Mar 06 2012

a(n) = 2*A099920(n+1)-A099920(n). - R. J. Mathar, Aug 23 2022

Mathematica

Table[Sum[LucasL[k]LucasL[n-k], {k, 0, n}], {n, 0, 40}] (* or *) LinearRecurrence[ {2, 1, -2, -1}, {4, 4, 13, 22}, 40] (* Harvey P. Dale, Mar 06 2012 *)

Crossrefs

Cf. A001629, A000032. Bisection: A203573 (even), 2*A203574 (odd).

Sequence in context: A323920 A005301 A358683 * A147824 A019081 A219454

Adjacent sequences: A099921 A099922 A099923 * A099925 A099926 A099927

Keyword

nonn,easy

Author

Ralf Stephan, Nov 01 2004

Status

approved