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

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, 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, 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

Offset: 0

Author

Hans Havermann, May 26 2002

Keywords

• nonn
• tabf

Comments

Row n has length 2n+1.

Also the coefficients of (x^2 + x + 1)^n mod 2. - Alan DenAdel, Mar 19 2014

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

Examples

			Triangle begins:
               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, 1, 1;
... - _Michel Marcus_, Mar 20 2014
		

References

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

Links

• Robert Price, Table of n, a(n) for n = 0..9999
• Rémy Sigrist, Representation of the first 2^10 rows of the table
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• S. Wolfram, A New Kind of Science
• Index to Elementary Cellular Automata
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A027907, A071052, A071053.

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

Programs

Mathematica

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 *)

PARI

rown(n) = Vec(lift((x^2 + x + 1)^n * Mod(1, 2))); \\ Michel Marcus, Mar 20 2014

Formula

a(n) = A027907(n) modulo 2. - Michel Marcus, Mar 20 2014

Extensions

Corrected by Hans Havermann, Jan 08 2012