

A169648


Number of cells that are turned from OFF to ON at stage n in Wolfram's 2D cellular automaton defined by Rule 942.


10



0, 1, 4, 4, 12, 8, 12, 12, 36, 28, 12, 12, 36, 28, 36, 36, 108, 100, 12, 12, 36, 28, 36, 36, 108, 92, 36, 36, 108, 84, 108, 108, 324, 340, 12, 12, 36, 28, 36, 36, 108, 92, 36, 36, 108, 84, 108, 108, 324, 316, 36, 36, 108, 84, 108, 108, 324, 276, 108, 108, 324
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OFFSET

1,3


COMMENTS

We work on the square grid. A cell is turned ON iff exactly one or four of its four neighbors is ON. Once a cell is ON it stays ON. At stage 1 all cells are OFF. At stage 0 a single cell is turned ON.
This sequence also arises from Rule 467 (New Kind of Science, page 173) if we count white cells, black cells in alternate generations.  N. J. A. Sloane, Feb 04 2015


REFERENCES

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 928.


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..549
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata


FORMULA

Added Apr 15 2010: The sequence is the sum of A147582(n) and 4 times A169689(4n), both of which have simple explicit formulas for the nth term. See those two entries for details.
The sequence is generated by the following recurrence (this is true, but unnecessarily complicated): Take a(0),...,a(7) as initial values.
For n >= 8, write n = 2^k + j with 0 <= j < 2^k. Then:
a(2^k)=3a(2^(k1))+3*2^(k1)8 (this is for j=0),
a(3*2^(k1))=3a(3*2^(k2))+2^(k+1)24 (this is for j=2^(k1)),
and otherwise
a(2^k+j)=a(2^(k1)+j) for 0 < j < 2^(k1)1,
a(2^k+j)=3a(2^(k1)+j) for 2^(k1) < j < 2^k.
The leading terms in the rows are essentially 4*A169651, and the "midpoints" of the rows are essentially 4*A169650.
See A169688, A169689 for a simpler (but equivalent) recurrence for this sequence.


EXAMPLE

May be written as a triangle:
0,
1,
4,12,
8,12,12,36,
28,12,12,36,28,36,36,108,
100,12,12,36,28,36,36,108,92,36,36,108,84,108,108,324,
340,12,12,36,28,36,36,108,92,36,36,108,84,108,108,324,316,36,...


CROSSREFS

Cf. A169649 (partial sums), A169650, A169651. See also A169688, A169689, A169689.
Sequence in context: A189545 A272990 A273645 * A273742 A169710 A269629
Adjacent sequences: A169645 A169646 A169647 * A169649 A169650 A169651


KEYWORD

nonn,tabf


AUTHOR

N. J. A. Sloane, Apr 07 2010, Apr 15 2010


STATUS

approved



