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A071044
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Number of ON cells at generation n of 1-D CA defined by Rule 22, starting with a single ON cell.
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3
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1, 3, 2, 6, 2, 6, 4, 12, 2, 6, 4, 12, 4, 12, 8, 24, 2, 6, 4, 12, 4, 12, 8, 24, 4, 12, 8, 24, 8, 24, 16, 48, 2, 6, 4, 12, 4, 12, 8, 24, 4, 12, 8, 24, 8, 24, 16, 48, 4, 12, 8, 24, 8, 24, 16, 48, 8, 24, 16, 48, 16, 48, 32, 96, 2, 6, 4, 12, 4, 12, 8, 24, 4, 12, 8, 24, 8, 24, 16, 48
(list;
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OFFSET
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0,2
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COMMENTS
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Number of 1's in n-th row of triangle in A071029.
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
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LINKS
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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.
Index entries for sequences related to cellular automata
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FORMULA
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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
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EXAMPLE
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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)
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A071029.
Sequence in context: A129354 A129628 A245691 * A077066 A324890 A180240
Adjacent sequences: A071041 A071042 A071043 * A071045 A071046 A071047
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KEYWORD
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nonn
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AUTHOR
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Hans Havermann, May 26 2002
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EXTENSIONS
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Better description from N. J. A. Sloane, Aug 15 2014
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STATUS
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approved
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