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A266072
Number of ON (black) cells in the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.
3
1, 1, 1, 5, 1, 9, 1, 13, 1, 17, 1, 21, 1, 25, 1, 29, 1, 33, 1, 37, 1, 41, 1, 45, 1, 49, 1, 53, 1, 57, 1, 61, 1, 65, 1, 69, 1, 73, 1, 77, 1, 81, 1, 85, 1, 89, 1, 93, 1, 97, 1, 101, 1, 105, 1, 109, 1, 113, 1, 117, 1, 121, 1, 125, 1, 129, 1, 133, 1, 137, 1, 141
OFFSET
0,4
COMMENTS
This sequence is A000012 and A016813 interspersed.
Also column 1 of A271343. - Omar E. Pol, Apr 06 2016
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
FORMULA
Conjectured g.f.: (1 + x - x^2 + 3*x^3)/(-1 + x^2)^2. - Michael De Vlieger, Dec 21 2015
Conjectures from Colin Barker, Dec 21 2015: (Start)
a(n) = n-(-1)^n*(n-1).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
EXAMPLE
From Michael De Vlieger, Dec 21 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row:
1 = 1
1 . . = 1
. . . 1 . = 1
1 1 1 1 . . 1 = 5
. . . . . . 1 . . = 1
1 1 1 1 1 1 1 . . 1 1 = 9
. . . . . . . . . 1 . . . = 1
1 1 1 1 1 1 1 1 1 1 . . 1 1 1 = 13
. . . . . . . . . . . . 1 . . . . = 1
1 1 1 1 1 1 1 1 1 1 1 1 1 . . 1 1 1 1 = 17
. . . . . . . . . . . . . . . 1 . . . . . = 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . 1 1 1 1 1 = 21
(End)
MATHEMATICA
rule = 3; 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}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Dec 20 2015
STATUS
approved