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
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
Programs
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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
Comments