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%I #18 Apr 16 2019 17:36:05
%S 1,3,6,11,17,25,34,45,57,71,86,103,121,141,162,185,209,235,262,291,
%T 321,353,386,421,457,495,534,575,617,661,706,753,801,851,902,955,1009,
%U 1065,1122,1181,1241,1303,1366,1431,1497,1565,1634,1705,1777,1851,1926
%N Total number of ON (black) cells after n iterations of the "Rule 157" 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="/A263807/b263807.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</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_, Jan 20 2016 and Apr 16 2019: (Start)
%F a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>3.
%F G.f.: (1+x+x^3) / ((1-x)^3*(1+x)).
%F (End)
%t rule=157; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) Table[Total[Take[nbc,k]],{k,1,rows}] (* Number of Black cells through stage n *)
%Y Cf. A263804.
%K nonn,easy
%O 0,2
%A _Robert Price_, Jan 17 2016
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