A203574 Bisection of A099924 (Lucas convolution); one half of the terms with odd arguments.

2, 11, 41, 137, 435, 1338, 4024, 11899, 34723, 100255, 286947, 815316, 2302286, 6466667, 18079805, 50343893, 139683219, 386328654, 1065440068, 2930780635, 8043131767, 22026515371, 60203886531, 164259660072, 447431169050, 1216927557323

Offset: 0

Author

Wolfdieter Lang, Jan 03 2012

Keywords

• nonn
• easy

Comments

The even part of this bisection of A099924 is found in A203573.

This is also the odd part of the bisection of A201207 (half-convolution of the Lucas sequence with itself). See a comment on A201204 for the definition of half-convolution of a sequence with itself. There the rule for the o.g.f. is given.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..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 (6,-11,6,-1).

Crossrefs

Cf. A000032, A000045, A099924, A203573, A201207.

Programs

Magma

I:=[2,11,41,137]; [n le 4 select I[n] else 6*Self(n-1) - 11*Self(n-2) + 6*Self(n-3) - Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 22 2017

Mathematica

CoefficientList[Series[(2-x-3x^2)/(1-3x+x^2)^2,{x,0,30}],x] (* or *) LinearRecurrence[{6,-11,6,-1},{2,11,41,137},30] (* Harvey P. Dale, Oct 12 2015 *)

PARI

x='x+O('x^30); Vec((2-x-3x^2)/(1-3x+x^2)^2) \\ G. C. Greubel, Dec 22 2017

Formula

a(n) = A099924(2*n+1)/2, n>=0.

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

a(n) = (3+2*n)*F(2*n) + (2+n)*F(2*n+1), with the Fibonacci numbers F(n)=A000045(n). From the partial fraction decomposition of the o.g.f. and the Fibonacci recurrence.

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4); a(0)=2, a(1)=11, a(2)=41, a(3)=137. - Harvey P. Dale, Oct 12 2015