A160722 Number of "ON" cells at n-th stage in a certain 2-dimensional cellular automaton based on Sierpinski triangles (see Comments for precise definition).
0, 1, 5, 9, 19, 23, 33, 43, 65, 69, 79, 89, 111, 121, 143, 165, 211, 215, 225, 235, 257, 267, 289, 311, 357, 367, 389, 411, 457, 479, 525, 571, 665, 669, 679, 689, 711, 721, 743, 765, 811, 821, 843, 865, 911, 933, 979, 1025, 1119, 1129, 1151, 1173, 1219, 1241
Offset: 0
Keywords
Examples
We start at round 0 with no polygons, a(0) = 0. At round 1 we turn ON the first triangle in each of the three Sierpinski triangles. After fusion we have a concave pentagon, so a(1) = 1. At round 2 we turn ON two triangles in each the three Sierpinski triangles. After fusions we have the concave pentagon and four triangles. So a(2) = 1 + 4 = 5.
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..9999
- 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.]
- Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 30.
- Omar E. Pol, Illustration if initial terms
- N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Programs
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Mathematica
a[0] = 0; a[1] = 1; a[n_] := a[n] = 2 a[Floor[#]] + a[Ceiling[#]] &[n/2]; Array[3 a[#] - 2 # &, 54, 0] (* Michael De Vlieger, Nov 01 2022 *)
Formula
a(n) = 3*A006046(n) - 2*n. - Max Alekseyev, Jan 21 2010
Extensions
Extended by Max Alekseyev, Jan 21 2010
Comments