A247650 Number of terms in expansion of f^n mod 2, where f = (1/x^2+1/x+1+x+x^2)*(1/y^2+1/y+1+y+y^2) mod 2.

1, 25, 25, 49, 25, 289, 49, 361, 25, 625, 289, 361, 49, 961, 361, 625, 25, 625, 625, 1225, 289, 3721, 361, 5041, 49, 1225, 961, 1681, 361, 5041, 625, 5929, 25, 625, 625, 1225, 625, 7225, 1225, 9025, 289, 7225, 3721, 5041, 361, 8281, 5041, 5929, 49, 1225

Offset: 0

Author

N. J. A. Sloane, Sep 25 2014

Keywords

• nonn

Comments

This is the number of cells that are ON after n generations in a two-dimensional cellular automaton defined by the odd-neighbor rule where the neighborhood consists of a 5X5 block of contiguous cells.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..200
• N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015

Crossrefs

Cf. A071053, A247647, A247648, A247649.

Programs

Python

import sympy
from operator import mul
from functools import reduce
x, y = sympy.symbols('x y')
f = ((1/x**2+1/x+1+x+x**2)*(1/y**2+1/y+1+y+y**2)).expand(modulus=2)
A247650_list, g = [1], 1
for n in range(1, 101):
    s = [int(d, 2) for d in bin(n)[2:].split('00') if d != '']
    g = (g*f).expand(modulus=2)
    if len(s) == 1:
        A247650_list.append(g.subs([(x, 1), (y, 1)]))
    else:
        A247650_list.append(reduce(mul, (A247650_list[d] for d in s)))
# Chai Wah Wu, Sep 25 2014

Formula

The values of a(n) for n in A247647 (or A247648) determine all the values, as follows. Parse the binary expansion of n into terms from A247647 separated by at least two zeros: m_1 0...0 m_2 0...0 m_3 ... m_r 0...0. Ignore any number (one or more) of trailing zeros. Then a(n) = a(m_1)*a(m_2)*...*a(m_r). For example, n = 37_10 = 100101_2 is parsed into 1.00.101, and so a(37) = a(1)*a(5) = 25*289 = 7225. This is a generalization of the Run Length Transform.