

A201207


Halfconvolution of sequence A000032 (Lucas) with itself.


2



4, 2, 7, 11, 27, 41, 84, 137, 270, 435, 826, 1338, 2488, 4024, 7353, 11899, 21461, 34723, 61960, 100255, 177344, 286947, 503892, 815316, 1422892, 2302286, 3996619, 6466667, 11173935, 18079805, 31114236
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OFFSET

0,1


COMMENTS

For the definition of the halfconvolution 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)=(2x)/(1xx^2) of A000032, and L2(x)= (47*xx^2)/((1+x)*(13*x+x^2)) the o.g.f. of A001254.
This leads to the o.g.f given in the formula section.
For the bisection of this sequence see A203570 and A203574.


LINKS

Table of n, a(n) for n=0..30.


FORMULA

a(n) = sum(L(k)*L(nk),n=0..floor(n/2)),n>=0, with the Lucas numbers L(n)=A000032(n).
O.g.f.: (42*x7*x^2+6*x^3x^4+3*x^5)/((13*x^2+x^4)*(1+x^2)*(1xx^2)). See a comment above.
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.


CROSSREFS

Cf. A000032, A000045, A201204, A203570, A203574.
Sequence in context: A019689 A072009 A257502 * A151890 A227352 A255140
Adjacent sequences: A201204 A201205 A201206 * A201208 A201209 A201210


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Jan 03 2012


STATUS

approved



