A203573 -id:A203573 - cp's OEIS frontend

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

Wolfdieter Lang, Jan 03 2012

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.

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).

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

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

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 *)

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

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

A099924 Self-convolution of Lucas numbers.

Original entry on oeis.org

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

Ralf Stephan, Nov 01 2004

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

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).

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

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 *)

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

A203570 Bisection of A201207 (half-convolution of the Lucas sequence A000032 with itself); even part.

Original entry on oeis.org

4, 7, 27, 84, 270, 826, 2488, 7353, 21461, 61960, 177344, 503892, 1422892, 3996619, 11173935, 31114236, 86328978, 238764238, 658478176, 1811322045, 4970928809, 13613135152, 37208048132, 101518052904, 276527670100, 752102592271

Offset: 0

Wolfdieter Lang, Jan 03 2012

nonn

The odd part of the bisection of A201207 is given in A203574.
See a comment on A201204 for the definition of the half-convolution of a sequence with itself, and the rule for the o.g.f.s of the bisection. Here the o.g.f. is (Lconve(x) + L2(x))/2, with the o.g.f. Lconve(x) = (4-11*x+11*x^2+x^3)/
(1-3*x+x^2)^2 of A203573 and the o.g.f. L2(x)= (4-7*x-x^2)/   ((1+x)*(1-3*x+x^2)) of A001254. This leads to the o.g.f. given in the formula section.

Cf. A201207, A203574.

a(n) = A201207(2*n), n>=0.
a(n) = (2*(4*n+6)*F(2*n+1)-4*(n+1)*F(2*n))/4 + (-1)^n, with the Fibonacci numbers F(n)=A000045(n).
O.g.f.: (4-13*x+4*x^3+12*x^2)/((1-3*x+x^2)^2*(1+x)). See a comment above.

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

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A099924 Self-convolution of Lucas numbers.

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A203570 Bisection of A201207 (half-convolution of the Lucas sequence A000032 with itself); even part.

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