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A266977 Number of ON (black) cells in the n-th iteration of the "Rule 78" elementary cellular automaton starting with a single ON (black) cell.

%I A266977 #33 Feb 16 2025 08:33:29
%S A266977 1,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,
%T A266977 16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,26,27,
%U A266977 27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35,35,36
%N A266977 Number of ON (black) cells in the n-th iteration of the "Rule 78" elementary cellular automaton starting with a single ON (black) cell.
%C A266977 Also, a(n-1) is the number of topologically inequivalent opening moves in the Sprouts game on n nodes [Browne]. - _Andrey Zabolotskiy_, Feb 12 2020
%C A266977 Also the number of symmetrically distinct faces in the 1 x 1 x (n+1) polycube. - _Eric W. Weisstein_, Sep 02 2022
%D A266977 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A266977 Robert Price, <a href="/A266977/b266977.txt">Table of n, a(n) for n = 0..1000</a>
%H A266977 Cameron B. Browne, <a href="https://eprints.qut.edu.au/101064/">Boundary Matching for Interactive Sprouts</a>, in: ACG 2015, pp. 147-159, LNCS 9525, Springer, doi:<a href="https://doi.org/10.1007/978-3-319-27992-3_14">10.1007/978-3-319-27992-3_14</a>.
%H A266977 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A266977 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A266977 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A266977 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%H A266977 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A266977 From _Colin Barker_, Jan 08 2016 and Apr 16 2019: (Start)
%F A266977 a(n) = (2*n+(-1)^n+7)/4 for n>0.
%F A266977 a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.
%F A266977 G.f.: (1+x-x^3) / ((1-x)^2*(1+x)). (End)
%F A266977 From _Stefano Spezia_, Aug 08 2021: (Start)
%F A266977 E.g.f.: ((4 + x)*cosh(x) + (3 + x)*sinh(x))/2 - 1.
%F A266977 a(n) = 2 + A004526(n) for n > 0. (End)
%t A266977 rule=78; 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 *) Table[Total[catri[[k]]],{k,1,rows}] (* Number of Black cells in stage n *)
%Y A266977 Cf. A004526, A266974.
%K A266977 nonn,easy
%O A266977 0,2
%A A266977 _Robert Price_, Jan 07 2016