A071036 - OEIS

%I #49 Feb 16 2025 08:32:46

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

%C Row n has length 2n+1.

%C Also the coefficients of (x^2 + x + 1)^n mod 2. - _Alan DenAdel_, Mar 19 2014

%C 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 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

%H Robert Price, <a href="/A071036/b071036.txt">Table of n, a(n) for n = 0..9999</a>

%H Rémy Sigrist, <a href="/A071036/a071036.png">Representation of the first 2^10 rows of the table</a>

%H Eric Weisstein's World of Mathematics, <a href="https://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="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F a(n) = A027907(n) modulo 2. - _Michel Marcus_, Mar 20 2014

%e Triangle begins:

%e                1;

%e             1, 1, 1;

%e          1, 0, 1, 0, 1;

%e       1, 1, 0, 1, 0, 1, 1;

%e    1, 0, 0, 0, 1, 0, 0, 0, 1;

%e 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1;

%e ... - _Michel Marcus_, Mar 20 2014

%t 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 (PARI) rown(n) = Vec(lift((x^2 + x + 1)^n * Mod(1, 2))); \\ _Michel Marcus_, Mar 20 2014

%Y Cf. A027907, A071052, A071053.

%Y This sequence, A038184 and A118110 are equivalent descriptions of the Rule 150 automaton.

%K nonn,tabf,changed

%O 0,1

%A _Hans Havermann_, May 26 2002

%E Corrected by _Hans Havermann_, Jan 08 2012