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A071044 Number of ON cells at generation n of 1-D CA defined by Rule 22, starting with a single ON cell. 3
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; graph; refs; listen; history; text; internal format)
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.

Index entries for sequences related to cellular automata

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

Cf. A071029.

Sequence in context: A129354 A129628 A245691 * A077066 A180240 A065228

Adjacent sequences:  A071041 A071042 A071043 * A071045 A071046 A071047

KEYWORD

nonn

AUTHOR

Hans Havermann, May 26 2002

EXTENSIONS

Better description from N. J. A. Sloane, Aug 15 2014

STATUS

approved

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Last modified August 21 12:01 EDT 2017. Contains 290864 sequences.