A265722 Number of ON (black) cells in the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.
1, 0, 1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, 24, 1, 28, 1, 32, 1, 36, 1, 40, 1, 44, 1, 48, 1, 52, 1, 56, 1, 60, 1, 64, 1, 68, 1, 72, 1, 76, 1, 80, 1, 84, 1, 88, 1, 92, 1, 96, 1, 100, 1, 104, 1, 108, 1, 112, 1, 116, 1, 120, 1, 124, 1, 128, 1, 132, 1, 136, 1, 140
Offset: 0
Examples
From _Michael De Vlieger_, Dec 14 2015: (Start)
First 12 rows, replacing zeros with "." for better visibility of ON cells, followed by the total number of 1's per row at right:
1 = 1
. . . = 0
. . 1 . . = 1
1 1 . . . 1 1 = 4
. . . . 1 . . . . = 1
1 1 1 1 . . . 1 1 1 1 = 8
. . . . . . 1 . . . . . . = 1
1 1 1 1 1 1 . . . 1 1 1 1 1 1 = 12
. . . . . . . . 1 . . . . . . . . = 1
1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 = 16
. . . . . . . . . . 1 . . . . . . . . . . = 1
1 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 1 = 20
(End)
References
- S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
Programs
-
Mathematica
rule = 1; rows = 30; Table[Total[Table[Take[CellularAutomaton[rule, {{1},0},rows-1,{All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]
Formula
Conjectures from Colin Barker, Dec 14 2015 and Apr 16 2019: (Start)
a(n) = 1/2*(-2*(-1)^n*n+2*n+3*(-1)^n-1).
a(n) = 2*a(n-2) - a(n-4) for n>3.
G.f.: (1-x^2+4*x^3) / ((1-x)^2*(1+x)^2).
(End)
a(n) = A019425(n), n>1. - R. J. Mathar, Jan 10 2016