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

Author

Robert Price, Jan 18 2016

Keywords

• nonn
• easy

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

Crossrefs

Cf. A000120, A006046, A006218, A070939, A071038, A080572, A267700.

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

Programs

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 *)

Formula

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

G.f.: (1/x)*(A(x)/x - (x+1)/(1-x)^2) where A(x) is the g.f. for A006046 (conjectured). - John Tyler Rascoe, Jul 08 2024