A079314 Number of first-quadrant cells (including the two boundaries) born at stage n of the Holladay-Ulam cellular automaton.

1, 2, 2, 4, 2, 4, 4, 10, 2, 4, 4, 10, 4, 10, 10, 28, 2, 4, 4, 10, 4, 10, 10, 28, 4, 10, 10, 28, 10, 28, 28, 82, 2, 4, 4, 10, 4, 10, 10, 28, 4, 10, 10, 28, 10, 28, 28, 82, 4, 10, 10, 28, 10, 28, 28, 82, 10, 28, 28, 82, 28, 82, 82, 244, 2, 4, 4, 10, 4, 10, 10, 28, 4, 10, 10, 28, 10, 28, 28, 82, 4

Offset: 0

Author

N. J. A. Sloane, Feb 12 2003

Keywords

• nonn
• easy

Comments

See the main entry for this CA, A147562, for further information.

When I first read the Singmaster MS in 2003 I misunderstood the definition of the CA. In fact once cells are ON they stay ON. The other version, when cells can change state from ON to OFF, is described in A079317. - N. J. A. Sloane, Aug 05 2009

The pattern has 4-fold symmetry; sequence just counts cells in one quadrant.

Examples

			From _Omar E. Pol_, Jul 18 2009: (Start)
If written as a triangle:
  1;
  2;
  2,4;
  2,4,4,10;
  2,4,4,10,4,10,10,28;
  2,4,4,10,4,10,10,28,4,10,10,28,10,28,28,82;
  2,4,4,10,4,10,10,28,4,10,10,28,10,28,28,82,4,10,10,28,10,28,28,82,10,28;...
Rows converge to A151712.
(End)
		

References

• D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
• Omar E. Pol, Illustration of initial terms (Overlapping squares) [From _Omar E. Pol_, Nov 20 2009]
• D. Singmaster, On the cellular automaton of Ulam and Warburton, 2003 [Cached copy, included with permission]
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015

Crossrefs

Cf. A147582, A079315, A079316, A079317, A079318, A079319, A151713, A139250.

Cf. A151712, A151922, A160407.

Programs

Mathematica

A079314list[nmax_]:=Join[{1},3^(DigitCount[Range[nmax],2,1]-1)+1];A079314list[100] (* Paolo Xausa, Jun 29 2023 *)

Formula

For n > 0, a(n) = 3^(A000120(n)-1) + 1.

For n > 0, a(n) = A147582(n)/4 + 1.

Partial sums give A151922. [Omar E. Pol, Nov 20 2009]

Extensions

Edited by N. J. A. Sloane, Aug 05 2009