A203570 Bisection of A201207 (half-convolution of the Lucas sequence A000032 with itself); even part.

4, 7, 27, 84, 270, 826, 2488, 7353, 21461, 61960, 177344, 503892, 1422892, 3996619, 11173935, 31114236, 86328978, 238764238, 658478176, 1811322045, 4970928809, 13613135152, 37208048132, 101518052904, 276527670100, 752102592271

Offset: 0

Author

Wolfdieter Lang, Jan 03 2012

Keywords

• nonn

Comments

The odd part of the bisection of A201207 is given in A203574.

See a comment on A201204 for the definition of the half-convolution of a sequence with itself, and the rule for the o.g.f.s of the bisection. Here the o.g.f. is (Lconve(x) + L2(x))/2, with the o.g.f. Lconve(x) = (4-11*x+11*x^2+x^3)/

(1-3*x+x^2)^2 of A203573 and the o.g.f. L2(x)= (4-7*x-x^2)/ ((1+x)*(1-3*x+x^2)) of A001254. This leads to the o.g.f. given in the formula section.

Crossrefs

Cf. A201207, A203574.

Formula

a(n) = A201207(2*n), n>=0.

a(n) = (2*(4*n+6)*F(2*n+1)-4*(n+1)*F(2*n))/4 + (-1)^n, with the Fibonacci numbers F(n)=A000045(n).

O.g.f.: (4-13*x+4*x^3+12*x^2)/((1-3*x+x^2)^2*(1+x)). See a comment above.