cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A250301 First differences of A250300.

Original entry on oeis.org

0, 3, 3, 6, 12, 12, 6, 12, 18, 18, 18, 18, 36, 36, 12, 24, 30, 18, 18, 24, 42, 54, 42, 36, 60, 66, 54, 48, 90, 90, 30, 54, 54, 18, 18, 24, 42, 54, 42, 42, 66, 78, 78, 78, 114, 150, 102, 72, 108
Offset: 0

Views

Author

Omar E. Pol, Jan 17 2015

Keywords

Comments

Number of cells turned "ON" at n-th stage of cellular automaton of A250300.
Compare A161645.

Examples

			Positive terms can be written as a triangle in which row lengths is A011782 as shown below:
3;
3;
6,  12;
12,  6, 12, 18;
18, 18, 18, 36, 36, 12, 24, 30;
18, 18, 24, 42, 54, 42, 36, 60, 66, 54, 48, 90, 90, 30, 54, 54;
18, 18, 24, 42, 54, 42, 42, 66, 78, 78, 78, 114, 150, 102, 72, 108, ...

Links

Crossrefs

Cf. A139251, A147582, A151724, A160121, A161645, A182633, A250300, A253771.

A161644 Number of ON states after n generations of cellular automaton based on triangles.

Original entry on oeis.org

0, 1, 4, 10, 16, 22, 34, 52, 64, 70, 82, 106, 136, 160, 190, 232, 256, 262, 274, 298, 328, 358, 400, 466, 532, 568, 598, 658, 742, 814, 892, 988, 1036, 1042, 1054, 1078, 1108, 1138, 1180, 1246, 1312, 1354, 1396, 1474, 1588, 1702, 1816, 1966, 2104, 2164, 2194
Offset: 0

Views

Author

David Applegate and N. J. A. Sloane, Jun 15 2009

Keywords

Comments

Analog of A151723 and A151725, but here we are working on the hexagonal net where each triangular cell has three neighbors (meeting along its edges). A cell is turned ON if exactly one of its three neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
There is a dual version where the triangular cells meet vertex-to-vertex. The counts are the same: the two versions are isomorphic. Reed (1974) uses the vertex-to-vertex version. See the two Sloane "Illustration" links below to compare the two versions.
It appears that a(n) is also the number of polytoothpicks added in a toothpick structure formed by V-toothpicks but starting with a Y-toothpick: a(n) = a(n-1)+(A182632(n)-A182632(n-1))/2. (Checked up to n=39.) - Omar E. Pol, Dec 07 2010 and R. J. Mathar, Dec 17 2010
It appears that the behavior is similar to A161206. - Omar E. Pol, Jan 15 2016
It would be nice to have a formula or recurrence.
If new triangles are required to always move outwards we get A295559 and A295560.
From Paul Cousin, May 23 2025: (Start)
This is ETA rule 242 (11110010 in binary):
-----------------------------------------------
|state of the cell |1|1|1|1|0|0|0|0|
|sum of the neighbors' states |3|2|1|0|3|2|1|0|
|cell's next state |1|1|1|1|0|0|1|0|
----------------------------------------------- (End)

References

  • R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Describes the dual structure where new triangles are joined at vertices rather than edges.]
  • S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962. See Example 3.

Links

Crossrefs

Cf. A151723, A151725, A147562, A161206, A161645, A139250, A160120, A161206, A182632, A182840, A250300, A295559, A295560.

Programs

  • PARI
    \\ See Links section.

Formula

a(n) = (A182632(n) - 1)/2, n >= 1. - Omar E. Pol, Mar 07 2013

Extensions

Edited by N. J. A. Sloane, Jan 10 2010 and Nov 27 2017

A253770 Number of ON states after n generations of cellular automaton based on triangles, with diamonds.

Original entry on oeis.org

0, 6, 24, 42, 96, 114, 168, 222, 348, 402, 456, 510, 636, 726, 852, 1014, 1320, 1482, 1536, 1590, 1716, 1806, 1932, 2094, 2400, 2598, 2724, 2886, 3192, 3498, 3840, 4254, 4956, 5442, 5568, 5622, 5748, 5838, 5964, 6126, 6432, 6630, 6756, 6918, 7224, 7530, 7872, 8286
Offset: 0

Views

Author

Omar E. Pol, Jan 11 2015

Keywords

Comments

Also 6 times the Y-toothpicks sequence A160120.
Explanation: consider the Y-toothpick structure of A160120, then replace every Y-toothpick with six ON cells forming a star with three rhombuses (or diamonds) that share only one vertex. Every diamond contains two triangular cells that share one edge.
The rules are the essentially the same as A160120.
An ON cell remains ON forever.
The sequence gives the number of triangular ON cells after the n-th stage.
A253771 (the first differences) give the number of triangular cells turned "ON" at the n-th stage.
A160120 (the Y-toothpick sequence) gives the number of stars in the structure after the n-th stage.
A160121 gives the number of stars added at the n-th stage.
A160167 gives the number of diamonds in the structure after the n-th stage.

Examples

			After one generation, the cellular automaton looks like a star or a flower with three petals as shown below:
.
.        /\
.       _\/_
.      /_/\_\
.
There are one star, three diamonds and six ON cells, so a(1) = 6.

Links

Crossrefs

Cf. A139250, A147562, A151723, A160120, A160157, A160167, A161644, A182632, A250300, A253771.

Formula

a(n) = 6*A160120(n) = 3*A160157(n) = 2*A160167(n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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