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A071036 Triangle read by rows giving successive states of cellular automaton generated by "Rule 150" when started with a single ON cell.

%I A071036 #49 Feb 16 2025 08:32:46
%S A071036 1,1,1,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0,1,1,1,1,0,1,1,1,0,1,
%T A071036 1,1,1,0,1,0,0,0,1,0,0,0,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,0,0,
%U A071036 0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0,0,1,1
%N A071036 Triangle read by rows giving successive states of cellular automaton generated by "Rule 150" when started with a single ON cell.
%C A071036 Row n has length 2n+1.
%C A071036 Also the coefficients of (x^2 + x + 1)^n mod 2. - _Alan DenAdel_, Mar 19 2014
%C A071036 The number of 0's in row n is A071052(n), and the number of 1's in row n is A071053(n). - _Michael Somos_, Jun 24 2018
%D A071036 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
%H A071036 Robert Price, <a href="/A071036/b071036.txt">Table of n, a(n) for n = 0..9999</a>
%H A071036 Rémy Sigrist, <a href="/A071036/a071036.png">Representation of the first 2^10 rows of the table</a>
%H A071036 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A071036 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A071036 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A071036 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A071036 a(n) = A027907(n) modulo 2. - _Michel Marcus_, Mar 20 2014
%e A071036 Triangle begins:
%e A071036                1;
%e A071036             1, 1, 1;
%e A071036          1, 0, 1, 0, 1;
%e A071036       1, 1, 0, 1, 0, 1, 1;
%e A071036    1, 0, 0, 0, 1, 0, 0, 0, 1;
%e A071036 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1;
%e A071036 ... - _Michel Marcus_, Mar 20 2014
%t A071036 T[ n_, k_] := T[n, k] = Which[k < 0 || k > 2 n, 0, n == k == 0, 1, True, Mod[ T[n - 1, k - 2] + T[n - 1, k - 1] + T[n - 1, k], 2]]; (* _Michael Somos_, Jun 24 2018 *)
%o A071036 (PARI) rown(n) = Vec(lift((x^2 + x + 1)^n * Mod(1, 2))); \\ _Michel Marcus_, Mar 20 2014
%Y A071036 Cf. A027907, A071052, A071053.
%Y A071036 This sequence, A038184 and A118110 are equivalent descriptions of the Rule 150 automaton.
%K A071036 nonn,tabf
%O A071036 0,1
%A A071036 _Hans Havermann_, May 26 2002
%E A071036 Corrected by _Hans Havermann_, Jan 08 2012