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
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
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
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jan 18 2016
STATUS
approved