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%I #21 Apr 16 2019 15:24:52
%S 0,3,7,10,18,21,33,36,52,55,75,78,102,105,133,136,168,171,207,210,250,
%T 253,297,300,348,351,403,406,462,465,525,528,592,595,663,666,738,741,
%U 817,820,900,903,987,990,1078,1081,1173,1176,1272,1275,1375,1378,1482
%N Total number of OFF (white) cells after n iterations of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H Robert Price, <a href="/A265724/b265724.txt">Table of n, a(n) for n = 0..999</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F Conjectures from _Colin Barker_, Dec 16 2015 and Apr 16 2019: (Start)
%F a(n) = 1/2*(n^2+(-1)^n*n+4*n-(-1)^n+1).
%F a(n) = 1/2*(n^2+5*n) for n even.
%F a(n) = 1/2*(n^2+3*n+2) for n odd.
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
%F G.f.: x*(3+4*x-3*x^2) / ((1-x)^3*(1+x)^2).
%F (End)
%e From _Michael De Vlieger_, Dec 14 2015: (Start)
%e First 12 rows, replacing ones with "." for better visibility of OFF cells, followed by the total number of 0's per row, and the running total up to that row:
%e . = 0 -> 0
%e 0 0 0 = 3 -> 3
%e 0 0 . 0 0 = 4 -> 7
%e . . 0 0 0 . . = 3 -> 10
%e 0 0 0 0 . 0 0 0 0 = 8 -> 18
%e . . . . 0 0 0 . . . . = 3 -> 21
%e 0 0 0 0 0 0 . 0 0 0 0 0 0 = 12 -> 33
%e . . . . . . 0 0 0 . . . . . . = 3 -> 36
%e 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 = 16 -> 52
%e . . . . . . . . 0 0 0 . . . . . . . . = 3 -> 55
%e 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 = 20 -> 75
%e . . . . . . . . . . 0 0 0 . . . . . . . . . . = 3 -> 78
%e (End)
%t rows = 53; Accumulate[Count[#, n_ /; n == 0] & /@ Table[Table[Take[CellularAutomaton[1, {{1}, 0}, rows - 1, {All, All}][[k]], {rows - k + 1, rows + k - 1}], {k, rows}][[k]], {k, 1, rows}]] (* _Michael De Vlieger_, Dec 14 2015 *)
%Y Cf. A265718, A265720, A265721.
%K nonn,easy
%O 0,2
%A _Robert Price_, Dec 14 2015
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