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A265225 Total number of ON (black) cells after n iterations of the "Rule 54" elementary cellular automaton starting with a single ON (black) cell. 1
1, 4, 6, 12, 15, 24, 28, 40, 45, 60, 66, 84, 91, 112, 120, 144, 153, 180, 190, 220, 231, 264, 276, 312, 325, 364, 378, 420, 435, 480, 496, 544, 561, 612, 630, 684, 703, 760, 780, 840, 861, 924, 946, 1012, 1035, 1104, 1128, 1200, 1225, 1300, 1326, 1404, 1431 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Take the first 2n positive integers and choose n of them such that their sum: a) is divisible by n, and b) is minimal. It seems their sum equals a(n). - Ivan N. Ianakiev, Feb 16 2019

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 08 2015 and Apr 20 2019: (Start)

a(n) = (n+1)*(2*n -(-1)^n +5)/4.

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.

G.f.: (1+3*x) / ((1-x)^3*(1+x)^2).

(End)

a(n) = n + 1 + (n+1) * floor((n+1)/2), conjectured. - Wesley Ivan Hurt, Dec 25 2016

EXAMPLE

From Michael De Vlieger, Dec 14 2015: (Start)

First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row, and the running total up to that row:

                        1                          =  1 ->  1

                      1 1 1                        =  3 ->  4

                    1 . . . 1                      =  2 ->  6

                  1 1 1 . 1 1 1                    =  6 -> 12

                1 . . . 1 . . . 1                  =  3 -> 15

              1 1 1 . 1 1 1 . 1 1 1                =  9 -> 24

            1 . . . 1 . . . 1 . . . 1              =  4 -> 28

          1 1 1 . 1 1 1 . 1 1 1 . 1 1 1            = 12 -> 40

        1 . . . 1 . . . 1 . . . 1 . . . 1          =  5 -> 45

      1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1        = 15 -> 60

    1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1      =  6 -> 66

  1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1    = 18 -> 84

1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1  =  7 -> 91

(End)

MAPLE

A265225:=n->1/4*(n+1)*(2*n-(-1)^n+5): seq(A265225(n), n=0..60); # Wesley Ivan Hurt, Dec 25 2016

MATHEMATICA

rule = 54; 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[Total /@ CellularAutomaton[54, {{1}, 0}, 52]]

CROSSREFS

Cf. A071030, A118108, A118109, A133872.

Sequence in context: A256241 A247632 A104236 * A122781 A153355 A310597

Adjacent sequences:  A265222 A265223 A265224 * A265226 A265227 A265228

KEYWORD

nonn,easy

AUTHOR

Robert Price, Dec 05 2015

STATUS

approved

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Last modified August 22 11:34 EDT 2019. Contains 326176 sequences. (Running on oeis4.)