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%I #46 Mar 16 2017 14:13:11
%S 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,
%T 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,
%U 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
%N Number of ON cells at generation n of 1-D CA defined by Rule 22, starting with a single ON cell.
%C Number of 1's in n-th row of triangle in A071029.
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
%H Robert Price, <a href="/A071044/b071044.txt">Table of n, a(n) for n = 0..1000</a>
%H Kari Eloranta, <a href="http://dx.doi.org/10.1088/0951-7715/6/6/010">Partially permutive cellular automata</a>, Nonlinearity 6.6 (1993): 1009. (Further information about Rule 22)
%H Peter Grassberger, <a href="http://dx.doi.org/10.1007/BF01033074">Long-range effects in an elementary cellular automaton</a>, Journal of Statistical Physics, 45.1-2 (1986): 27-39. (Further information about Rule 22)
%H A. J. Macfarlane, <a href="http://www.damtp.cam.ac.uk/user/ajm/Papers2016/GFsForCAsOfEvenRuleNo.ps">Generating functions for integer sequences defined by the evolution of cellular automata...</a>
%H N. J. A. Sloane, <a href="/A071044/a071044.png">Illustration of first 21 generations</a>
%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.
%H S. Wolfram, <a href="http://dx.doi.org/10.1103/RevModPhys.55.601">Statistical mechanics of cellular automata</a>, Rev. Mod. Phys., 55 (1983), 601--644.
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F 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
%F G.f. = (1+3*x)*Prod_{k >= 1} (1+2*x^(2^k)). - _N. J. A. Sloane_, Aug 09 2014
%e From _Michael De Vlieger_, Oct 05 2015: (Start)
%e 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:
%e 1 1
%e 3 1 1 1
%e 2 1 . . . 1
%e 6 1 1 1 . 1 1 1
%e 2 1 . . . . . . . 1
%e 6 1 1 1 . . . . . 1 1 1
%e 4 1 . . . 1 . . . 1 . . . 1
%e 12 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1
%e 2 1 . . . . . . . . . . . . . . . 1
%e (End)
%t ArrayPlot[CellularAutomaton[22, {{1}, 0}, 20]] (* _N. J. A. Sloane_, Aug 15 2014 *)
%t Total /@ CellularAutomaton[22, {{1}, 0}, 80] (* _Michael De Vlieger_, Oct 05 2015 *)
%Y Cf. A071029.
%K nonn
%O 0,2
%A _Hans Havermann_, May 26 2002
%E Better description from _N. J. A. Sloane_, Aug 15 2014
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