

A247649


Number of terms in expansion of f^n mod 2, where f = 1/x^2 + 1/x + 1 + x + x^2 mod 2.


11



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, 95, 71, 85, 25, 113, 77, 103
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OFFSET

0,2


COMMENTS

This is the number of cells that are ON after n generations in a onedimensional cellular automaton defined by the oddneighbor rule where the neighborhood consists of 5 contiguous cells.


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, 2015
Index entries for sequences related to cellular automata


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;
5,25,25,35,25,85,35,95,17,85,61,71,19,91,71,77,7,35,35,49,31,107,41,121,19,95,71,85,25,113,77,103;
...
(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:
A247649_list.append(reduce(mul, (A247649_list[d] for d in s)))
# Chai Wah Wu, Sep 25 2014


CROSSREFS

Cf. A071053, A247647, A247648, A253085, A255490.
Sequence in context: A088201 A195380 A139261 * A252655 A021646 A231589
Adjacent sequences: A247646 A247647 A247648 * A247650 A247651 A247652


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Sep 25 2014


STATUS

approved



