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A263511 Total number of ON (black) cells after n iterations of the "Rule 155" elementary cellular automaton starting with a single ON (black) cell.

%I A263511 #70 Jul 19 2025 00:18:28
%S A263511 1,3,6,12,19,29,40,54,69,87,106,128,151,177,204,234,265,299,334,372,
%T A263511 411,453,496,542,589,639,690,744,799,857,916,978,1041,1107,1174,1244,
%U A263511 1315,1389,1464,1542,1621,1703,1786,1872,1959,2049,2140,2234,2329,2427
%N A263511 Total number of ON (black) cells after n iterations of the "Rule 155" elementary cellular automaton starting with a single ON (black) cell.
%C A263511 Conjecture: it appears that these number also describe the number of numerically different eigenvalues of the triangular grid graphs with an odd number of edges on each side of the outer triangle, starting with |eigenvalues(T_3)|=6, |eigenvalues(T_5)|=12, etc. (checked up to |eigenvalues(T_13)|=54). - _Michael Terhoeven_, Jul 12 2025
%D A263511 Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A263511 Robert Price, <a href="/A263511/b263511.txt">Table of n, a(n) for n = 0..1000</a>
%H A263511 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A263511 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGridGraph.html">Triangular Grid Graph</a>
%H A263511 Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A263511 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A263511 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A263511 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A263511 G.f.: (1+x+2*x^3)/((1-x)^3*(1+x)). - _Vincenzo Librandi_, Jan 18 2016
%F A263511 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>3. - _Vincenzo Librandi_, Jan 18 2016
%t A263511 rule=155; 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 *)
%t A263511 LinearRecurrence[{2, 0, -2, 1}, {1, 3, 6, 12}, 50] (* _Vincenzo Librandi_, Jan 18 2016 *)
%o A263511 (Magma) I:=[1,3,6,12]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..60]]; // _Vincenzo Librandi_, Jan 18 2016
%Y A263511 Cf. A263243.
%K A263511 nonn,easy
%O A263511 0,2
%A A263511 _Robert Price_, Jan 17 2016