A283642 - OEIS

%I #27 Sep 10 2021 22:02:18

%S 1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691,87381,

%T 174763,349525,699051,1398101,2796203,5592405,11184811,22369621,

%U 44739243,89478485,178956971,357913941,715827883,1431655765,2863311531,5726623061,11453246123

%N Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 678", based on the 5-celled von Neumann neighborhood.

%C Initialized with a single black (ON) cell at stage zero.

%C Similar to A001045.

%C It is not difficult to prove that one has indeed a(n) = round(4*2^n/3) = A001045(n+2) for all n. The proof as well as the growth of the pattern is nearly identical to that of the toothpick sequence A139250. - _M. F. Hasler_, Feb 13 2020

%C The decimal representations of the n-th interval of elementary cellular automata rules 28 and 156 (see A266502 and A266508) generate this sequence. - _Karl V. Keller, Jr._, Sep 03 2021

%H Robert Price, <a href="/A283642/b283642.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A283642/a283642.tmp.txt">Diagrams of first 20 stages</a>

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>, Wolfram Media, 2002; p. 170.

%H Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</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_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).

%F From _Colin Barker_, Mar 14 2017: (Start)

%F G.f.: (1 + 2*x) / ((1 + x)*(1 - 2*x)).

%F a(n) = (2^(n+2) - 1) / 3 for n even.

%F a(n) = (2^(n+2) + 1) / 3 for n odd.

%F a(n) = a(n-1) + 2*a(n-2) for n>1.

%F (End)

%F I.e., a(n) = A001045(n+2) = A154917(n+2) = A167167(n+2) = |A077925(n+1)| = A328284(n+5) = round(4*2^n/3), cf. comments. - _M. F. Hasler_, Feb 13 2020

%F E.g.f.: (4*exp(2*x) - exp(-x))/3. - _Stefano Spezia_, Feb 13 2020

%F a(n) = floor((4*2^n + 1)/3). - _Karl V. Keller, Jr._, Sep 03 2021

%t CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];

%t code = 678; stages = 128;

%t rule = IntegerDigits[code, 2, 10];

%t g = 2 * stages + 1; (* Maximum size of grid *)

%t a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)

%t ca = a;

%t ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];

%t PrependTo[ca, a];

%t (* Trim full grid to reflect growth by one cell at each stage *)

%t k = (Length[ca[[1]]] + 1)/2;

%t ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];

%t Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]

%o (Python) print([(4*2**n + 1)//3 for n in range(50)]) # _Karl V. Keller, Jr._, Sep 03 2021

%Y Cf. A283641, A266502, A266508, A086893, A001045.

%K nonn,easy

%O 0,2

%A _Robert Price_, Mar 12 2017