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

0,2

Comments

See A204449 for the exponential half-convolution of A000032 (Lucas). The present sequence gives the numbers to be added to A204449 to obtain the corresponding (full) exponential convolution A203579.

Links

Table of n, a(n) for n=0..25.

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.

Example

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.

Crossrefs

Cf. A000032, A203579, A204449.

Sequence in context: A024310 A064516 A001441 * A037261 A239234 A192707

Adjacent sequences: A204447 A204448 A204449 * A204451 A204452 A204453

Keyword

nonn

Author

Wolfdieter Lang, Jan 16 2012

Status

approved