%N Number of ON cells after n generations of two-dimensional automaton based on knight moves (see Comments for definition; here a cell is turned ON if 1 or 2 neighbors are ON).
%C The cells are the squares of the standard square grid.
%C Cells are either OFF or ON, once they are ON they stay ON forever.
%C Each cell has 8 neighbors, the cells that are a knight's move away.
%C We begin in generation 0 with a single ON cell.
%C A cell is turned ON at generation n+1 if it has either one or two ON neighbor at generation n.
%C Since cells stay ON, an equivalent definition is that a cell is turned ON at generation n+1 if it has one or two neighbors that has been turned ON at some earlier generation.
%C This sequence is a variant of A319018.
%C This is another knight's-move version of the Ulam-Warburton cellular automaton (see A147562).
%C The structure has dihedral D_8 symmetry (quarter-turn rotations plus reflections), so A322055 is a multiple of 8.
%H Rémy Sigrist, <a href="/A322055/b322055.txt">Table of n, a(n) for n = 0..1000</a>
%H Rémy Sigrist, <a href="/A322055/a322055_1.png">Illustration of the structure at stage 255</a>
%H N. J. A. Sloane, <a href="/A322055/a322055.png">Illustration of a(0) to a(5).</a>
%F Conjectures from _Colin Barker_, Dec 22 2018: (Start)
%F G.f.: (1 + 8*x + 32*x^2 + 32*x^3 + 70*x^4 + 24*x^5 + 72*x^6 + 49*x^8 - 8*x^10 + 16*x^11 - 8*x^12) / ((1 - x)^3*(1 + x)^2*(1 + x^2)^2).
%F a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n>8.
%Y Cf. A139250, A147562, A319018, A319019, A322056.
%A _N. J. A. Sloane_, Dec 21 2018
%E More terms from _Rémy Sigrist_, Dec 22 2018