A113859 Expansion of (7-14*x+6*x^2)/((1-x)*(2*x^2-4*x+1)); related to the binomial transform of Pell numbers A000129 (see formula and comment for A007070).

7, 21, 69, 233, 793, 2705, 9233, 31521, 107617, 367425, 1254465, 4283009, 14623105, 49926401, 170459393, 581984769, 1987020289, 6784111617, 23162405889, 79081400321, 270000789505, 921840357377, 3147359850497, 10745758687233

Offset: 0

Author

Creighton Dement, Jan 25 2006

Keywords

• easy
• nonn

Comments

If g.f. (x^6+5*x^4+6*x^2+1)/(x^7+6*x^5+10*x^3+4*x) is expanded, where (x^6+5*x^4+6*x^2+1) and (x^7+6*x^5+10*x^3+4*x) are the 7th and 8th Fibonacci polynomials, respectively, the sequence: [0, 7/8, 0, -21/16, 0, 69/32, 0, -233/64, 0, 793/128, 0, -2705/256, ] is returned. (a(n)) is seen to be the numerators of the bisection of this sequences, apart from signs.

Crossrefs

Cf. A007070, A007052, A003480, A111567, A000129.

Programs

Maple

with(combinat, fibonacci): seq(fibonacci(i, x), i=1..15); [[generates sequence of Fibonacci polynomials]]

Formula

a(n+1) - a(n) = A007070(n+2), a(n) - 2*a(n+1) + a(n+2) = A007052(n+3) (Number of order consecutive partitions of n), a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n) = A003480(n+4), a(n+2) - a(n) = A111567(n+3)