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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). 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 (list; graph; refs; listen; history; text; internal format)
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 n-th 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), 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

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

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Last modified December 9 19:36 EST 2022. Contains 358703 sequences. (Running on oeis4.)