OFFSET
0,2
COMMENTS
Number of ON cells at generation n of 1-D CA defined by Rule 126, starting with a single ON cell. - N. J. A. Sloane, Aug 09 2014
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
LINKS
Robert Price, Table of n, a(n) for n = 0..1000
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata...
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601--644.
FORMULA
a(n) = 2^(1+wt(n)) unless n is of the form 2^i-1 in which case we must subtract 1, where wt = A000120. - N. J. A. Sloane, Aug 09 2014
G.f.: 2*Prod_{k=0..oo} (1+2*x^(2^k)) - Sum_k=0..oo} x^(2^k-1). - N. J. A. Sloane, Aug 09 2014
EXAMPLE
[Contribution from Omar E. Pol, Dec 11 2010] (Start)
May be arranged in blocks of sizes 1, 1, 2, 4, 8, 16, 32, ...:
1,
3,
4, 7,
4, 8, 8, 15,
4, 8, 8, 16, 8, 16, 16, 31,
4, 8, 8, 16, 8, 16, 16, 32, 8, 16, 16, 32, 16, 32, 32, 63,
Last terms of rows give positive terms of A000225.
(End)
MATHEMATICA
a[n_] := 2^(DigitCount[n, 2, 1]+1) - Boole[IntegerQ[Log[2, n+1]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 02 2018, from 2nd formula *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Hans Havermann, May 26 2002
STATUS
approved