

A160722


Number of "ON" cells at nth stage in a certain 2dimensional cellular automaton based on Sierpinski triangles (see Comments for precise definition).


8



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
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OFFSET

0,3


COMMENTS

This cellular automata is formed by the concatenation of three Sierpinski triangles, starting from a central vertex. Adjacent polygons are fused. The ON cells are triangles, but we only count after fusion. The sequence gives the number of polygons at the nth round.
If instead we start from four Sierpinski triangles we get A160720.


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), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.]
HsienKuei Hwang, Svante Janson, and TsungHsi Tsai, Identities and periodic oscillations of divideandconquer 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


FORMULA

a(n) = 3*A006046(n)  2*n.  Max Alekseyev, Jan 21 2010


EXAMPLE

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.


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 *)


CROSSREFS

A160723 gives the first differences.
Cf. A139250, A160720.
Sequence in context: A226663 A023521 A113805 * A255652 A061202 A235799
Adjacent sequences: A160719 A160720 A160721 * A160723 A160724 A160725


KEYWORD

nonn


AUTHOR

Omar E. Pol, May 25 2009, Jan 03 2010


EXTENSIONS

Extended by Max Alekseyev, Jan 21 2010


STATUS

approved



