OFFSET
0,2
COMMENTS
Number of 1's in n-th row of triangle in A071029.
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
Kari Eloranta, Partially permutive cellular automata, Nonlinearity 6.6 (1993): 1009. (Further information about Rule 22)
Peter Grassberger, Long-range effects in an elementary cellular automaton, Journal of Statistical Physics, 45.1-2 (1986): 27-39. (Further information about Rule 22)
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata...
N. J. A. Sloane, Illustration of first 21 generations
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
If the binary expansion of n is b_{r-1} b_{r-2} ... b_2 b_1 b_0, then a(n) = 3^b_0 * Prod_{i=1..r-1} 2^b_i = 2^wt(n) if n is even, or (3/2)*2^wt(n) if n is odd (cf. A000120). - N. J. A. Sloane, Aug 09 2014
G.f. = (1+3*x)*Prod_{k >= 1} (1+2*x^(2^k)). - N. J. A. Sloane, Aug 09 2014
EXAMPLE
From Michael De Vlieger, Oct 05 2015: (Start)
First 8 rows, replacing "0" with "." for better visibility of ON cells, total of ON cells in each row to the left of the diagram:
1 1
3 1 1 1
2 1 . . . 1
6 1 1 1 . 1 1 1
2 1 . . . . . . . 1
6 1 1 1 . . . . . 1 1 1
4 1 . . . 1 . . . 1 . . . 1
12 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1
2 1 . . . . . . . . . . . . . . . 1
(End)
MATHEMATICA
ArrayPlot[CellularAutomaton[22, {{1}, 0}, 20]] (* N. J. A. Sloane, Aug 15 2014 *)
Total /@ CellularAutomaton[22, {{1}, 0}, 80] (* Michael De Vlieger, Oct 05 2015 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Hans Havermann, May 26 2002
EXTENSIONS
Better description from N. J. A. Sloane, Aug 15 2014
STATUS
approved