A267610 Total number of OFF (white) cells after n iterations of the "Rule 182" elementary cellular automaton starting with a single ON (black) cell.

0, 0, 2, 2, 4, 6, 12, 12, 14, 16, 22, 24, 30, 36, 50, 50, 52, 54, 60, 62, 68, 74, 88, 90, 96, 102, 116, 122, 136, 150, 180, 180, 182, 184, 190, 192, 198, 204, 218, 220, 226, 232, 246, 252, 266, 280, 310, 312, 318, 324, 338, 344, 358, 372, 402, 408, 422, 436

Offset

0,3

Comments

From Gus Wiseman, Mar 30 2019: (Start)

It appears that a(n) is also the number of increasing binary-containment pairs of distinct positive integers up to n + 1. A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second. For example, the a(2) = 2 through a(8) = 14 pairs are:

{1,3} {1,3} {1,3} {1,3} {1,3} {1,3} {1,3}

{2,3} {2,3} {1,5} {1,5} {1,5} {1,5} {1,5}

{2,3} {2,3} {1,7} {1,7} {1,7}

{4,5} {2,6} {2,3} {2,3} {1,9}

{4,5} {2,6} {2,6} {2,3}

{4,6} {2,7} {2,7} {2,6}

{3,7} {3,7} {2,7}

{4,5} {4,5} {3,7}

{4,6} {4,6} {4,5}

{4,7} {4,7} {4,6}

{5,7} {5,7} {4,7}

{6,7} {6,7} {5,7}

{6,7}

{8,9}

(End)

References

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Robert Price, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Formula

Conjecture: a(n) = A267700(n) - n. - Gus Wiseman, Mar 30 2019

Mathematica

rule=182; rows=20; ca=CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]], {rows-k+1, rows+k-1}], {k, 1, rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]], {k, 1, rows}]; (* Number of Black cells in stage n *) nwc=Table[Length[catri[[k]]]-nbc[[k]], {k, 1, rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc, k]], {k, 1, rows}] (* Number of White cells through stage n *)

Crossrefs

Cf. A000120, A006218, A070939, A071038, A080572, A267700.

Cf. A325101, A325103, A325104, A325106, A325109, A325110, A325123, A325124.

Sequence in context: A195204 A318847 A228892 * A336940 A291365 A154779

Adjacent sequences: A267607 A267608 A267609 * A267611 A267612 A267613

Keyword

nonn,easy

Author

Robert Price, Jan 18 2016

Status

approved