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A071028 Triangle read by rows giving successive states of cellular automaton generated by "Rule 50".

%I A071028 #57 Feb 16 2025 08:32:46
%S A071028 1,1,0,1,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,
%T A071028 0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,
%U A071028 1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0
%N A071028 Triangle read by rows giving successive states of cellular automaton generated by "Rule 50".
%C A071028 Row n has length 2n+1.
%C A071028 Rules #50, #58, #114, #122, #178, #179, #186, #242, #250 all give rise to this sequence.
%C A071028 The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - _Jeremy Gardiner_, Mar 16 2003
%D A071028 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
%H A071028 Robert Price, <a href="/A071028/b071028.txt">Table of n, a(n) for n = 0..9999</a>
%H A071028 C. J. Glasby, S. P. Glasby, and F. Pleijel, <a href="http://dx.doi.org/10.1098/rspb.2008.0418">Worms by number</a>, Proc. Roy. Soc. B, Proc. Biol. Sci. 275 (1647) (2008) 2071-2076.
%H A071028 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule250.html">Rule 250</a>
%H A071028 Michael Williams, <a href="https://doi.org/10.13140/RG.2.2.29146.31686">Collatz conjecture: an order isomorphic recursive machine</a>, ResearchGate (2024). See pp. 8, 13.
%H A071028 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A071028 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A071028 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A071028 a(n) = n - 1 + floor(sqrt(n)) - 2*Sum_{k=1..n-1} a(k) for n >= 1. - _Benoit Cloitre_, Jan 24 2013
%F A071028 a(n) = A071797(n) (mod 2). - _Boris Putievskiy_, Jul 24 2013
%F A071028 a(n) = (1+(-1)^(Sum_{k=1..floor(n/2)} floor((n-k)/k)))/2. - _Wesley Ivan Hurt_, Dec 25 2020
%e A071028 Triangle begins:
%e A071028 1;
%e A071028 1, 0, 1;
%e A071028 1, 0, 1, 0, 1;
%e A071028 1, 0, 1, 0, 1, 0, 1;
%e A071028 1, 0, 1, 0, 1, 0, 1, 0, 1;
%e A071028 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
%e A071028 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
%e A071028 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
%e A071028 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
%e A071028 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
%e A071028 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
%e A071028 - _Philippe Deléham_, Mar 23 2014
%t A071028 rows = 10; ca = CellularAutomaton[50, {{1}, 0}, rows-1]; Flatten[ Table[ca[[k, rows-k+1 ;; rows+k-1]], {k, 1, rows}]] (* _Jean-François Alcover_, May 24 2012 *)
%Y A071028 Cf. A071797.
%K A071028 nonn,tabf
%O A071028 0,1
%A A071028 _Hans Havermann_, May 26 2002