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

0,1

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).

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A144841 A203245 A121244 * A070778 A260267 A128241

Adjacent sequences: A203571 A203572 A203573 * A203575 A203576 A203577

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 03 2012

Status

approved