A385493 Number of distinct states in Conway's Game of Life acting on a (2n+1) X (2n+1) toroidal grid starting with (x,y) turned on if and only if x-n + (y-n)*i is a Gaussian prime.

1, 1, 1, 6, 9, 4, 4, 5, 14, 12, 17, 5, 8, 19, 15, 34, 20, 21, 19, 77, 52, 29, 58, 39, 27, 27, 68, 31, 27, 27, 27, 70, 49, 129, 83, 43, 153, 40, 82, 128, 60, 457, 436, 79, 99, 71, 71, 178, 125, 281, 121, 121, 94, 231, 94, 94, 385, 122, 94, 94, 175, 306, 156

Offset: 0

Author

Luke Bennet, Jun 30 2025

Keywords

• nonn

Examples

			For a(3), the sequence of Conway's Game of Life is
 | . o . o . o . | . o . o . o . | . o . . . o . |
 | o . o . o . o | o . . . . . o | o o o o o o o |
 | . o o . o o . | . . o . o . . | . o . . . o . |
 | o . . . . . o | o . . . . . o | . o . . . o . |
 | . o o . o o . | . . o . o . . | . o . . . o . |
 | o . o . o . o | o . . . . . o | o o o o o o o |
 | . o . o . o . | . o . o . o . | . o . . . o . |
  (generation 1)  (generation 2)  (generation 3)
 | . . . o . . . | . . o o o . . | . o . . . o . |
 | . . . o . . . | . . o o o . . | o . . . . . o |
 | . . . o . . . | o o . o . o o | . . . . . . . |
 | o o o . o o o | o o o . o o o | . . . . . . . |
 | . . . o . . . | o o . o . o o | . . . . . . . |
 | . . . o . . . | . . o o o . . | o . . . . . o |
 | . . . o . . . | . . o o o . . | . o . . . o . |
  (generation 4)  (generation 5)  (generation 6)
Every generation after 6 is identical to generation 6, so this sequence has 6 unique states. Thus, a(3) = 6.
		

Crossrefs

Cf. A055025.

Programs

Python

import torch
import numpy as np
def prime_mask(limit):
    is_prime = torch.ones(limit + 1, dtype=torch.bool)
    is_prime[:2] = False
    for i in range(2, int(limit**0.5) + 1):
        if is_prime[i]:
            is_prime[i*i : limit+1 : i] = False
    return is_prime
def Gauss_primes(N):
    A, B = torch.meshgrid(torch.arange(-N, N+1), torch.arange(-N, N+1))
    norm = A**2 + B**2
    is_prime = prime_mask(2*N**2)
    mask = (A != 0) & (B != 0) & is_prime[norm]
    axis_mask = ((A == 0) ^ (B == 0))
    axis_val = (A + B).abs()
    axis_mask &= is_prime[axis_val] & ((axis_val % 4) == 3)
    return mask | axis_mask
def update(G):
    shifts = [(1,0),(1,1),(0,1),(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)]
    neighbors = sum(torch.roll(G, shifts=shift, dims=(0,1)) for shift in shifts)
    return (G & ((neighbors == 2) | (neighbors == 3))) | (~G & (neighbors == 3))
def a(n):
    if n == 0 or n == 1:
        return 1
    G = Gauss_primes(n).to("cuda").to(torch.uint8)
    seen, step = set(), 0
    while True:
        flat = G.flatten().to("cpu").numpy()
        key = bytes(np.packbits(flat))
        if key in seen:
            return step
        seen.add(key)
        G = update(G)
        step += 1