cp's OEIS Frontend

A322055 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).

%I A322055 #21 Dec 22 2018 03:55:34
%S A322055 1,9,41,73,145,185,321,385,577,649,881,993,1297,1401,1729,1889,2305,
%T A322055 2441,2865,3073,3601,3769,4289,4545,5185,5385,6001,6305,7057,7289,
%U A322055 8001,8353,9217,9481,10289,10689,11665,11961,12865,13313,14401,14729,15729,16225
%N A322055 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 A322055 The cells are the squares of the standard square grid.
%C A322055 Cells are either OFF or ON, once they are ON they stay ON forever.
%C A322055 Each cell has 8 neighbors, the cells that are a knight's move away.
%C A322055 We begin in generation 0 with a single ON cell.
%C A322055 A cell is turned ON at generation n+1 if it has either one or two ON neighbor at generation n.
%C A322055 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 A322055 This sequence is a variant of A319018.
%C A322055 This is another knight's-move version of the Ulam-Warburton cellular automaton (see A147562).
%C A322055 The structure has dihedral D_8 symmetry (quarter-turn rotations plus reflections), so A322055 is a multiple of 8.
%H A322055 Rémy Sigrist, <a href="/A322055/b322055.txt">Table of n, a(n) for n = 0..1000</a>
%H A322055 Rémy Sigrist, <a href="/A322055/a322055_1.png">Illustration of the structure at stage 255</a>
%H A322055 N. J. A. Sloane, <a href="/A322055/a322055.png">Illustration of a(0) to a(5).</a>
%F A322055 Conjectures from _Colin Barker_, Dec 22 2018: (Start)
%F A322055 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 A322055 a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n>8.
%F A322055 (End)
%Y A322055 Cf. A139250, A147562, A319018, A319019, A322056.
%K A322055 nonn
%O A322055 0,2
%A A322055 _N. J. A. Sloane_, Dec 21 2018
%E A322055 More terms from _Rémy Sigrist_, Dec 22 2018