A320030 Automaton sum similar to A102376 but using mod 3 instead of mod 2.

1, 4, 13, 4, 16, 52, 13, 52, 121, 4, 16, 52, 16, 64, 208, 52, 208, 484, 13, 52, 121, 52, 208, 484, 121, 484, 1093, 4, 16, 52, 16, 64, 208, 52, 208, 484, 16, 64, 208, 64, 256, 832, 208, 832, 1936, 52, 208, 484, 208, 832, 1936, 484, 1936, 4372, 13, 52, 121, 52

Offset

1,2

Comments

The automaton that generates this sequence operates on a grid of cells c(i,j). The cells have three possible values, 0, 1, and 2. The next generation in the CA is calculated by applying the following rule to each cell: c(i,j) = ( c(i+1,j-1) + c(i+1,j+1) + c(i-1,j-1) + c(i-1,j+1) ) mod 3.

Start with a single cell with a value of 1, with all other cells set to 0. For each generation, the term in this sequence c(n) is the aggregate values of all cells in the grid for each discrete generation of the automaton (i.e., not cumulative over multiple generations).

The cellular automaton that generates this sequence has been empirically observed to repeat the number of active cells (4 in this case) if the iteration number N is a power of the modulus + 1. The modulus in this case is 3.

This has been observed to occur with any prime modulus and any starting pattern of cells. I'm picking this particular implementation because it's the same as the one used in A102376.

Counting the active (nonzero) cells instead of taking the sum also creates a different but related sequence. This sequence is the sum of each iteration, and cells in this automaton have values 0, 1, or 2. Only for mod 2 are both the sum and active cell counts the same.

Links

Table of n, a(n) for n=1..58.

Nathan M Epstein, Animation of CA

Formula

a(3^n) = A096053(n).

Prog

(Python)
import numpy as np
from scipy import signal
frameSize = 301
filter = [[0, 1, 0], [1, 0, 1], [0, 1, 0]] # this defines the CA neighborhood
frame  = np.zeros((frameSize, frameSize))
frame[frameSize/2, frameSize/2] = 1
mod = 3
sequence = []
for j in range(140):
    frame = signal.convolve2d(frame, filter, mode='same')
    frame = np.mod(frame, mod)
    sequence.append(np.sum(frame.reshape(1, -1)))

Crossrefs

Cf. A096053.

Cf. A102376 (mod 2), A320100 (mod 5).

Sequence in context: A265327 A130650 A170865 * A191509 A218356 A249120

Adjacent sequences: A320027 A320028 A320029 * A320031 A320032 A320033

Keyword

nonn

Author

Nathan M Epstein, Dec 10 2018

Status

approved