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).

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

Offset: 0

Author

N. J. A. Sloane, Dec 21 2018

Keywords

• nonn

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's-move version of the Ulam-Warburton cellular automaton (see A147562).

The structure has dihedral D_8 symmetry (quarter-turn 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).

Crossrefs

Cf. A139250, A147562, A319018, A319019, A322056.

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(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n>8.

(End)

Extensions

More terms from Rémy Sigrist, Dec 22 2018