A099256 - OEIS

%I #24 Mar 12 2024 13:20:11

%S 3,8,9,23,24,61,63,160,165,419,432,1097,1131,2872,2961,7519,7752,

%T 19685,20295,51536,53133,134923,139104,353233,364179,924776,953433,

%U 2421095,2496120,6338509,6534927,16594432,17108661,43444787,44791056,113739929,117264507,297775000,307002465,779585071

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

%C 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.

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

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

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).

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

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

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

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

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

%t LinearRecurrence[{0,3,0,-1},{3,8,9,23},40] (* _Harvey P. Dale_, Apr 22 2012 *)

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

%K nonn,easy

%O 0,1

%A _Creighton Dement_, Oct 18 2004

%E Definition corrected, extended. - _R. J. Mathar_, Nov 13 2008