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%I #13 Jan 23 2020 03:26:12
%S 4,2,7,11,27,41,84,137,270,435,826,1338,2488,4024,7353,11899,21461,
%T 34723,61960,100255,177344,286947,503892,815316,1422892,2302286,
%U 3996619,6466667,11173935,18079805,31114236
%N Half-convolution of sequence A000032 (Lucas) with itself.
%C For the definition of the half-convolution of a sequence with itself see a comment on A201204. There the rule for the o.g.f. is given. Here the o.g.f. is (L(x)^2 + L2(x^2))/2, with the o.g.f. L(x)=(2-x)/(1-x-x^2) of A000032, and L2(x) = (4-7*x-x^2)/((1+x)*(1-3*x+x^2)) the o.g.f. of A001254. This leads to the o.g.f given in the formula section.
%C For the bisection of this sequence see A203570 and A203574.
%F a(n) = Sum_{n=0..floor(n/2)} (L(k)*L(n-k), n >= 0, with the Lucas numbers L(n)=A000032(n).
%F O.g.f.: (4-2*x-7*x^2+6*x^3-x^4+3*x^5)/((1-3*x^2+x^4)*(1+x^2)*(1-x-x^2)). See a comment above.
%F a(n) = (1/4)*(2*(2*n+5+(-1)^n)*F(n+1)-(2*n+3+(-1)^n)*F(n)) +(i^n+(-i)^n)/2, n >= 0, with the Fibonacci numbers F(n)=A000045(n) and the imaginary unit i=sqrt(-1). From the partial fraction decomposition of the o.g.f. and the Fibonacci recurrence.
%Y Cf. A000032, A000045, A201204, A203570, A203574.
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Jan 03 2012
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