[seqfan] Re: back to quaternions...

[Richard Mathar]

On behalf of http://list.seqfan.eu/pipermail/seqfan/2010-February/003580.html ,

F[+-e](n), essentially A054878(n),  = 0,3,6,21,60,183,546,1641,4920,14763,.. (n>=1)
   = (3^n+3*(-1)^n)/4
has g.f. 3*x^2/((1+x)*(1-3*x)) = -1 + ( 3/(1+x)+1/(1-3*x)) /4

F[e](n) = 0, 0, 3, 15, 30, 78, 273, 861, 2460, .. (n>=1)
        = 3 * ( 0, 0, 1,5,10,26,91,287,820,2420,7381, 22265,...)
        = A054878(n)/2 + (sign alternating, aerated A000244(n))/2
has g.f. 3*x^3*(1+3*x)/((1+x)*(1-3*x)*(1+3*x^2))

and the difference
F[-e](n) = 3*(0,1,1,2,10,35,91,260,820,2501,...) = F[+-e](n) - F[e](n)
     = 0,3,3,6,30,105,273,780,2460,7503,...
     = A054878(n)/2 - (sign alternating, aerated A000244(n))/2
has g.f. 3*x^2*(1-x)/((1+x)*(1-3*x)*(1+3*x^2))
         = ( 3/(1+x) + 1/(1-3*x) -4/(1+3*x^2) )/8  ;