

A189007


Number of ON cells after n generations of the 2D cellular automaton described in the comments.


4



1, 4, 8, 16, 16, 32, 32, 64, 32, 64, 64, 128, 64, 128, 128, 256, 64, 128, 128, 256, 128, 256, 256, 512, 128, 256, 256, 512, 256, 512, 512, 1024, 128, 256, 256, 512, 256, 512, 512, 1024, 256, 512, 512, 1024, 512, 1024, 1024, 2048, 256, 512, 512, 1024, 512, 1024, 1024, 2048, 512, 1024, 1024, 2048, 1024, 2048, 2048, 4096, 256, 512, 512, 1024, 512
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OFFSET

1,2


COMMENTS

The cells are the squares of the standard infinite square grid. All cells are initially OFF and a single cell is turned ON at generation 1. At subsequent generations a cell is ON if and only if exactly one East/West neighbor was ON or exactly one North/South neighbor was ON (or BOTH of those conditions) in the previous generation.
The equivalent Mathematica cellular automaton is obtained with neighborhood weights {{0,1,0},{3,0,3},{0,1,0}}}, rule number 186, and initial configuration {{1}}.
Also sequence generated by Rule 84 with neighborhood weights {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}.  Robert Price, Mar 11 2016


LINKS

Table of n, a(n) for n=1..69.
John W. Layman, Graphs of the automaton for generations 115


FORMULA

It appears that this sequence is the limit of the following process. Start with {1,4} and repeatedly perform this set of operations: (1) select the second half H of the sequence; (2) append twice the terms of H, then (3) append four times the terms of H. This gives {1,4} > {1,4,8,16} > {1,4,8,16,16,32,32,64} > {1,4,8,16,16,32,32,64,32,64,64,128,64,128,128,256} > ... This has been verified for the first 150 terms.
Comment from N. J. A. Sloane, Jul 21 2014 (Start): It is not difficult to show that the preceding conjecture is correct. In fact one can give an explicit formula for the nth term. At generation n >= 2, the configuration of ON cells consists of a set of concentric diamonds (see the illustration). The sizes of the diamonds are given by the (n2)nd term of A245191. Let N = A245191(n2) = Sum_{i>=0} b_i*2^i. Then the ON cells form a set of diamonds with edgelengths i+2 for each b_i = 1. The ith diamond contains 4*(i+1) ON cells, and the total number of ON cells is therefore a(n) = 4*Sum_i (i+1)*b_i. The b_i are given explicitly in A245191.
For example, if n=11, N = A245191(9) = 544 = 2^5 + 2^9, so b_5 = b_9 = 1, there are two diamonds, of side lengths 7 and 11, containing a total of 4*(6+10) = 64 = a(11) ON cells. (End)


MATHEMATICA

ca = CellularAutomaton[{186, {2, {{0, 1, 0}, {3, 0, 3}, {0, 1, 0}}}, {1, 1}}, {{{1}}, 0}, 501, 50]; Table[Total[ca[[n]], 2], {n, 1, 50}]


CROSSREFS

Cf. A164982, A165345, A245191.
Sequence in context: A312763 A312764 A166634 * A242349 A309521 A072603
Adjacent sequences: A189004 A189005 A189006 * A189008 A189009 A189010


KEYWORD

nonn


AUTHOR

John W. Layman, Apr 15 2011


STATUS

approved



