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A203574 Bisection of A099924 (Lucas convolution); one half of the members with odd arguments. 5
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 (list; graph; refs; listen; history; text; internal format)
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, 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(0)=2, a(1)=11, a(2)=41, a(3)=137, a(n) = 6*a(n-1)-11*a(n-2)+6*a(n-3)-a(n-4). - 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

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Last modified January 23 01:37 EST 2021. Contains 340384 sequences. (Running on oeis4.)