OFFSET
0,2
COMMENTS
This is the number of cells that are ON after n generations in a one-dimensional cellular automaton defined by the odd-neighbor rule where the neighborhood consists of 5 contiguous cells.
a(n) is also the number of odd entries in row n of A035343. - Leon Rische, Feb 02 2023
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
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) = 5*17 = 85. This is a generalization of the Run Length Transform.
EXAMPLE
The first few generations are:
..........X..........
........XXXXX........
......X.X.X.X.X......
....XX..X.X.X..XX.... (f^3)
..X...X...X...X...X..
XXXX.XXX.XXX.XXX.XXXX
...
f^3 mod 2 = x^6 + x^5 + x^2 + 1/x^2 + 1/x^5 + 1/x^6 + 1 has 7 terms, so a(3) = 7.
From Omar E. Pol, Mar 02 2015: (Start)
Also, written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
5;
5, 7;
5,17, 7,19;
5,25,17,19, 7,31,19,25;
5,25,25,35,17,61,19,71, 7,35,31,41,19,71,25,77;
5,25,25,35,25,85,35,95,17,85,61,71,19,91,71,77,7,35,35,49,31,107,41,121,19, ...
(End)
It follows from the Generalized Run Length Transform result mentioned in the comments that in each row the first quarter of the terms (and no more) are equal to 5 times the beginning of the sequence itself. It cannot be said that the rows converge (in any meaningful sense) to five times the sequence. - N. J. A. Sloane, Mar 03 2015
PROG
(Python)
import sympy
from functools import reduce
from operator import mul
x = sympy.symbols('x')
f = 1/x**2+1/x+1+x+x**2
A247649_list, g = [1], 1
for n in range(1, 1001):
s = [int(d, 2) for d in bin(n)[2:].split('00') if d != '']
g = (g*f).expand(modulus=2)
if len(s) == 1:
A247649_list.append(g.subs(x, 1))
else:
# Chai Wah Wu, Sep 25 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Sep 25 2014
STATUS
approved