%I #27 Sep 08 2022 08:46:01
%S 2,11,41,137,435,1338,4024,11899,34723,100255,286947,815316,2302286,
%T 6466667,18079805,50343893,139683219,386328654,1065440068,2930780635,
%U 8043131767,22026515371,60203886531,164259660072,447431169050,1216927557323
%N Bisection of A099924 (Lucas convolution); one half of the terms with odd arguments.
%C The even part of this bisection of A099924 is found in A203573.
%C 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.
%H G. C. Greubel, <a href="/A203574/b203574.txt">Table of n, a(n) for n = 0..1000</a>
%H É. Czabarka, R. Flórez, and L. Junes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Florez/florez12.html">A Discrete Convolution on the Generalized Hosoya Triangle</a>, Journal of Integer Sequences, 18 (2015), #15.1.6.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).
%F a(n) = A099924(2*n+1)/2, n>=0.
%F O.g.f.: (2-x-3*x^2)/(1-3*x+x^2)^2.
%F 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.
%F 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
%t 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 *)
%o (PARI) x='x+O('x^30); Vec((2-x-3x^2)/(1-3x+x^2)^2) \\ _G. C. Greubel_, Dec 22 2017
%o (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
%Y Cf. A000032, A000045, A099924, A203573, A201207.
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Jan 03 2012
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