

A322055


Number of ON cells after n generations of twodimensional automaton based on knight moves (see Comments for definition; here a cell is turned ON if 1 or 2 neighbors are ON).


3



1, 9, 41, 73, 145, 185, 321, 385, 577, 649, 881, 993, 1297, 1401, 1729, 1889, 2305, 2441, 2865, 3073, 3601, 3769, 4289, 4545, 5185, 5385, 6001, 6305, 7057, 7289, 8001, 8353, 9217, 9481, 10289, 10689, 11665, 11961, 12865, 13313, 14401, 14729, 15729, 16225
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OFFSET

0,2


COMMENTS

The cells are the squares of the standard square grid.
Cells are either OFF or ON, once they are ON they stay ON forever.
Each cell has 8 neighbors, the cells that are a knight's move away.
We begin in generation 0 with a single ON cell.
A cell is turned ON at generation n+1 if it has either one or two ON neighbor at generation n.
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.
This sequence is a variant of A319018.
This is another knight'smove version of the UlamWarburton cellular automaton (see A147562).
The structure has dihedral D_8 symmetry (quarterturn rotations plus reflections), so A322055 is a multiple of 8.


LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Rémy Sigrist, Illustration of the structure at stage 255
N. J. A. Sloane, Illustration of a(0) to a(5).


FORMULA

Conjectures from Colin Barker, Dec 22 2018: (Start)
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).
a(n) = a(n1) + 2*a(n4)  2*a(n5)  a(n8) + a(n9) for n>8.
(End)


CROSSREFS

Cf. A139250, A147562, A319018, A319019, A322056.
Sequence in context: A034925 A159754 A045804 * A198943 A000451 A000437
Adjacent sequences: A322052 A322053 A322054 * A322056 A322057 A322058


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Dec 21 2018


EXTENSIONS

More terms from Rémy Sigrist, Dec 22 2018


STATUS

approved



