A169648 Number of cells that are turned from OFF to ON at stage n in Wolfram's 2-D cellular automaton defined by Rule 942.

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

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, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

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 n-th 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^(k-1))+3*2^(k-1)-8 (this is for j=0),

a(3*2^(k-1))=3a(3*2^(k-2))+2^(k+1)-24 (this is for j=2^(k-1)),

and otherwise

a(2^k+j)=a(2^(k-1)+j) for 0 < j < 2^(k-1)-1,

a(2^k+j)=3a(2^(k-1)+j) for 2^(k-1) < 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: A326993 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