A071044 Number of ON cells at generation n of 1-D CA defined by Rule 22, starting with a single ON cell.
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
Offset: 0
Keywords
Examples
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)
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
Crossrefs
Cf. A071029.
Programs
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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 *)
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
Extensions
Better description from N. J. A. Sloane, Aug 15 2014
Comments