A266591 - OEIS

%I #31 Jul 01 2023 15:13:29

%S 1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,

%T 1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,

%U 1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0

%N Middle column of the "Rule 37" elementary cellular automaton starting with a single ON (black) cell.

%C Also the number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UD and DU. The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude. This sequence is periodic with a pre-period of length 2 (namely 1, 1) and a period of length 2 (namely 1, 0). It can be shown (via induction) that the middle column of rule 37 is eventually periodic, too. - _Valerie Roitner_, Apr 02 2020

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

%H Robert Price, <a href="/A266591/b266591.txt">Table of n, a(n) for n = 0..1000</a>

%H Andrei Asinowski, Cyril Banderier, and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, preprint, 2019.

%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>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1).

%F G.f.: (1+t-t^3)/((1-t)*(1+t)). - _Valerie Roitner_, Jan 13 2020

%F A signed form of the sequence, starting 1, 1, 1, 0, -1, 0, 1, ..., is s(n) = signum([x^n] B(x)), where B(x) = x/(1 - exp(-x)) is the e.g.f. of the Bernoulli numbers with B(1) = 1/2. - _Peter Luschny_, Jan 23 2023

%e For n=2k we only have one allowed excursion, namely (UD)^k. For n=1, however, the excursion H is allowed (since we have not attained length two or more, no forbidden consecutive step sequence has appeared yet). - _Valerie Roitner_, Jan 13 2020

%t rule=37; 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[catri[[k]][[k]],{k,1,rows}] (* Keep only middle cell from each row *)

%Y Cf. A059841, A266588.

%K nonn,easy,walk

%O 0

%A _Robert Price_, Jan 01 2016