A160723 -id:A160723 - cp's OEIS frontend

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

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

nonn

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.

			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.

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

A160723 gives the first differences.

Cf. A139250, A160720.

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 *)

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

Extended by Max Alekseyev, Jan 21 2010

A160721 First differences of A160720.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 12, 28, 4, 12, 12, 28, 12, 28, 28, 60, 4, 12, 12, 28, 12, 28, 28, 60, 12, 28, 28, 60, 28, 60, 60, 124, 4, 12, 12, 28, 12, 28, 28, 60, 12, 28, 28, 60, 28, 60, 60, 124, 12, 28, 28, 60, 28, 60, 60, 124, 28, 60, 60, 124, 60, 124, 124, 252, 4, 12, 12, 28, 12, 28, 28

Offset: 1

Omar E. Pol, May 25 2009, May 29 2009

This sequence is related to the Sierpinski triangle and to Gould's sequence A001316. - Omar E. Pol, Jul 23 2009

When written as a irregular triangle in which row lengths are A011782 it appears that right border gives A173033. - Omar E. Pol, Mar 20 2013

			From _Omar E. Pol_, Mar 20 2013 (Start):
Triangle begins:
1;
4;
4,12;
4,12,12,28;
4,12,12,28,12,28,28,60;
4,12,12,28,12,28,28,60,12,28,28,60,28,60,60,124;
4,12,12,28,12,28,28,60,12,28,28,60,28,60,60,124,12,28,28,60,28,60,60,124,28,60,60,124,60,124,124,252;
(End)

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.]
David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000120, A001316, A038573, A139250, A139251, A160720, A160722, A160723.

a(1)=1. Observation: It appears that a(n) = 4*A038573(n-1), n>1. [From Omar E. Pol, Jul 23 2009]. This formula is correct! - N. J. A. Sloane, Jan 23 2016

More terms from R. J. Mathar, Jul 14 2009

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions