A099256 Expansion of g.f. (3-x)*(1+3*x+x^2)/((1-x-x^2)*(1+x-x^2)).

3, 8, 9, 23, 24, 61, 63, 160, 165, 419, 432, 1097, 1131, 2872, 2961, 7519, 7752, 19685, 20295, 51536, 53133, 134923, 139104, 353233, 364179, 924776, 953433, 2421095, 2496120, 6338509, 6534927, 16594432, 17108661, 43444787, 44791056, 113739929, 117264507, 297775000, 307002465, 779585071

Offset: 0

Author

Creighton Dement, Oct 18 2004

Keywords

• nonn
• easy

Comments

One of two sequences involving the Lucas/Fibonacci numbers. This sequence consists of pairs of numbers more or less close to each other with "jumps" in between pairs.

a(n+3) + a(n) - a(n+2) appears to be mysteriously connected with a(n+1).

Both this sequence and A099255 were created using "Floretion dynamical symmetries" (see link for further details).

Links

• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Crossrefs

Cf. A000045, A099255, A000032, A055273 (bisection), A097134 (bisection).

Programs

Mathematica

LinearRecurrence[{0,3,0,-1},{3,8,9,23},40] (* Harvey P. Dale, Apr 22 2012 *)

Formula

a(2n+2) - a(2n+1) = Fibonacci(2n-1).

A099255(n)/2 - a(n)/2 = (-1)^n*A000032(n)

a(0) = 3, a(1) = 8, a(2) = 9, a(3) = 23, a(n+4) = 3a(n+2) - a(n).

a(2n) = A022086(2n+2), a(2n+1) = A022097(2n+2).

a(n) = A013655(n+2)-A061084(n+1).

Extensions

Definition corrected, extended. - R. J. Mathar, Nov 13 2008