A265429 Total number of ON (black) cells after n iterations of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.
1, 3, 5, 9, 13, 18, 23, 30, 37, 45, 53, 63, 73, 84, 95, 108, 121, 135, 149, 165, 181, 198, 215, 234, 253, 273, 293, 315, 337, 360, 383, 408, 433, 459, 485, 513, 541, 570, 599, 630, 661, 693, 725, 759, 793, 828, 863, 900, 937, 975, 1013, 1053, 1093, 1134
Offset: 0
Examples
From _Michael De Vlieger_, Dec 09 2015: (Start) First 12 rows, replacing "0" with ".", ignoring "0" outside of range of 1's for better visibility of ON cells, followed by total number of ON cells per row, and running total up to that row: 1 = 1 -> 1 1 1 = 2 -> 3 1 . 1 = 2 -> 5 1 1 1 1 = 4 -> 9 1 1 1 . 1 = 4 -> 13 1 1 . 1 1 1 = 5 -> 18 1 . 1 1 1 . 1 = 5 -> 23 1 1 1 1 . 1 1 1 = 7 -> 30 1 1 1 . 1 1 1 . 1 = 7 -> 37 1 1 . 1 1 1 . 1 1 1 = 8 -> 45 1 . 1 1 1 . 1 1 1 . 1 = 8 -> 53 1 1 1 1 . 1 1 1 . 1 1 1 = 10 -> 63 1 1 1 . 1 1 1 . 1 1 1 . 1 = 10 -> 72 (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..999
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
Crossrefs
Cf. A118174.
Programs
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Mathematica
rule = 188; rows = 30; Table[Total[Take[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}],k]],{k,1,rows}] Accumulate[Count[#, n_ /; n == 1] & /@ CellularAutomaton[188, {{1}, 0}, 53]] (* Michael De Vlieger, Dec 09 2015 *)
Formula
Conjectures from Colin Barker, Dec 09 2015 and Apr 16 2019: (Start)
a(n) = (1/16)*(6*n^2 + 24*n - 3*(-1)^n + 2*(-i)^n + 2*i^n + 15) where i = sqrt(-1).
G.f.: (1 + x + 2*x^3 - x^4) / ((1-x)^3*(1+x)*(1+x^2)).
(End)
Conjecture: the sequence consists of all numbers k > 0 such that floor(sqrt(8*(k+1)/3)) != floor(sqrt(8*k/3)). - Gevorg Hmayakyan, Sep 01 2019