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

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, 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, 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

Offset

0,1

Comments

Row n has length 2n+1.

Rules #50, #58, #114, #122, #178, #179, #186, #242, #250 all give rise to this sequence.

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003

References

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

Links

Robert Price, Table of n, a(n) for n = 0..9999

C. J. Glasby, S. P. Glasby, and F. Pleijel, Worms by number, Proc. Roy. Soc. B, Proc. Biol. Sci. 275 (1647) (2008) 2071-2076.

Eric Weisstein's World of Mathematics, Rule 250

Michael Williams, Collatz conjecture: an order isomorphic recursive machine, ResearchGate (2024). See pp. 8, 13.

Stephen Wolfram, A New Kind of Science

Index to Elementary Cellular Automata

Index entries for sequences related to cellular automata

Formula

a(n) = n - 1 + floor(sqrt(n)) - 2*Sum_{k=1..n-1} a(k) for n >= 1. - Benoit Cloitre, Jan 24 2013

a(n) = A071797(n+1) (mod 2). - Boris Putievskiy, Jul 24 2013

a(n) = (1+(-1)^(Sum_{k=1..floor(n/2)} floor((n-k)/k)))/2. - Wesley Ivan Hurt, Dec 25 2020

a(n) = (1 + (-1)^A028392(n))/2. - Ridouane Oudra, Mar 02 2026

Example

Triangle begins:
  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, 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, 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, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
  - Philippe Deléham, Mar 23 2014

Mathematica

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 *)

Crossrefs

Cf. A071797, A028392.

Sequence in context: A014240 A014471 A230002 * A286987 A011635 A015752

Adjacent sequences: A071025 A071026 A071027 * A071029 A071030 A071031

Keyword

nonn,tabf,easy

Author

Hans Havermann, May 26 2002

Status

approved