This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A265724 #26 Feb 16 2025 08:33:28
%S A265724 0,3,7,10,18,21,33,36,52,55,75,78,102,105,133,136,168,171,207,210,250,
%T A265724 253,297,300,348,351,403,406,462,465,525,528,592,595,663,666,738,741,
%U A265724 817,820,900,903,987,990,1078,1081,1173,1176,1272,1275,1375,1378,1482
%N A265724 Total number of OFF (white) cells after n iterations of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.
%D A265724 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A265724 Robert Price, <a href="/A265724/b265724.txt">Table of n, a(n) for n = 0..999</a>
%H A265724 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A265724 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A265724 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A265724 Conjectures from _Colin Barker_, Dec 16 2015 and Apr 16 2019: (Start)
%F A265724 a(n) = 1/2*(n^2+(-1)^n*n+4*n-(-1)^n+1).
%F A265724 a(n) = 1/2*(n^2+5*n) for n even.
%F A265724 a(n) = 1/2*(n^2+3*n+2) for n odd.
%F A265724 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
%F A265724 G.f.: x*(3+4*x-3*x^2) / ((1-x)^3*(1+x)^2).
%F A265724 (End)
%F A265724 Apparently, a(n) = A267049(n) + 4*floor(n/2) - 1 for n>1. - _Hugo Pfoertner_, Jun 21 2024
%e A265724 From _Michael De Vlieger_, Dec 14 2015: (Start)
%e A265724 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 A265724 . = 0 -> 0
%e A265724 0 0 0 = 3 -> 3
%e A265724 0 0 . 0 0 = 4 -> 7
%e A265724 . . 0 0 0 . . = 3 -> 10
%e A265724 0 0 0 0 . 0 0 0 0 = 8 -> 18
%e A265724 . . . . 0 0 0 . . . . = 3 -> 21
%e A265724 0 0 0 0 0 0 . 0 0 0 0 0 0 = 12 -> 33
%e A265724 . . . . . . 0 0 0 . . . . . . = 3 -> 36
%e A265724 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 = 16 -> 52
%e A265724 . . . . . . . . 0 0 0 . . . . . . . . = 3 -> 55
%e A265724 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 = 20 -> 75
%e A265724 . . . . . . . . . . 0 0 0 . . . . . . . . . . = 3 -> 78
%e A265724 (End)
%t A265724 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 A265724 Cf. A265718, A265720, A265721, A267049.
%K A265724 nonn,easy
%O A265724 0,2
%A A265724 _Robert Price_, Dec 14 2015