A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata

By
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger

.pdf   .ps   .tex

Posted: March 5, 2015

The present little article (accompanied by a not-so-little Maple package), uses methods similar to those in this article to answer some questions considered in this article .

# Maple Package

• CAcount, To generate enumeration automata for counting the number of ON Cells in Cellular Automata given by odd rules, and extensions to mod p

# Some Input and Output files for the Maple package CAcount

• If you want to see an article regarding A246039 and A246038, followed by an article regarding A253069 and A253070

the input   yields the output

• If you want to see an article regarding mod 3 analogs of A246039 and A246038, followed by an article regarding mod 3 analogs A253069 and A253070, keeping track of the individuality of the coefficients

the input   yields the output

• If you want to see an article regarding mod 5 analogs of A246039 and A246038, followed by an article regarding mod 5 analogs A253069 and A253070, keeping track of the individuality of the coefficients

the input   yields the output

• If you want to see some random examples of 1-dimensional cellular automata given by odd rules and generalizations with mod 3 and mod 5

the input   yields the output

• If you want to see some random examples of 1-dimensional cellular automata given by odd rules and generalizations with mod 3 and mod 5, but keeping track of the individuality of the coefficients

the input   yields the output

• If you want to see mod 3 analogs of A246039 and A246038, and of A253069 and A253070

the input   yields the output

• If you want to see all sequences

P(x)n mod 2 evaluated at x=1

for all polynomials of degree ≤ 7 with at least three monomials

the input   yields the output

• If you want to see all sequences

P(x)n mod 3 evaluated at x=1

for all polynomials of degree ≤ 7 with at least three monomials where the individuality of the coeeficients is kept

the input   yields the output

• If you want to see all sequences

P(x,y)n mod 2 evaluated at x=1

for all polynomials P(x,y) whose support is contained in {0,1,2}x{0,1,2}

the input   yields the output

• If you want to see all sequences

P(x,y)n mod 3, where the individuality of the coefficients is kept

for all polynomials P(x,y) whose support is contained in {0,1,2}x{0,1,2}

the input   yields the output

• If you want to see an article about the cellular automata generated by the 3D Moore neighborhood, the subject of OEIS sequences A246031 and A246032

the input   yields the output

• More impressively, and time-consuming, is an article about the cellular automata generated by the 4D Moore neighborhood,

the input   yields the output

[Because this output file is so long, for your convenience, we extract the generating function for the sparse subsequence, whose denominator has degree 221 (and numerator, degree 220) in Maple input format, and called it Moore4, in this output file]
[Note, these sequences were entered in the OEIS as A255477 and A244478]

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger