A204450 2*A203579 - A204449. Difference between the exponential convolution of A000032 (Lucas) with itself and the corresponding exponential half-convolution.
0, 2, 6, 17, 30, 177, 417, 1857, 4302, 19457, 47731, 203777, 509769, 2134017, 5462701, 22347777, 58104062, 234029057, 616919457, 2450784257, 6533317815, 25664946177, 69085604341, 268766806017, 729558799305, 2814562533377
Offset: 0
Keywords
Examples
With A000032 = {2, 1, 3, 4, 7, 11, ...}
a(4) = 4*4*1 + 1*7*2 = 30.
a(5) = 10*4*3 + 5*7*1 + 1*11*2 = 177.
Formula
a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=floor(n/2)+1..n), n>=0,
with the Lucas numbers L(n)=A000032(n). For n=0 this is 0.
E.g.f.: exp(x)*(cosh(sqrt(5)*x)+1) - (BesselI(0,2*phi*x) + BesselI(0,2*(phi-1)*x) + 2*BesselI(0,2*I*x))/2. Compare this with the e.g.f. of A204449, where phi and BesselI are explained.
Bisection: a(2*k) = (2^(2*k)-binomial(2*k,k))*L(2*k)/2 + 1 - ((-1)^k)*binomial(2*k,k), a(2*k+1) = 2^(2*k)*L(2*k+1)+ 1 = A204449(2*k+1), k>=0.
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