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

0,2

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

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

Example

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)

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 *)

Crossrefs

Cf. A118174.

Sequence in context: A032635 A036713 A260733 * A356254 A122248 A024403

Adjacent sequences: A265426 A265427 A265428 * A265430 A265431 A265432

Keyword

nonn,easy

Author

Robert Price, Dec 08 2015

Status

approved