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.

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A169709 Total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 1006".

Original entry on oeis.org

1, 5, 9, 21, 29, 41, 61, 89, 101, 113, 133, 161, 189, 233, 309, 377, 397, 409, 429, 457, 485, 529, 605, 673, 709, 753, 821, 905, 1005, 1169, 1413, 1569, 1605, 1617, 1637, 1665, 1693, 1737, 1813, 1881, 1917, 1961, 2029, 2113, 2213, 2377, 2621, 2777, 2829, 2873, 2941
Offset: 0

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Author

N. J. A. Sloane, Apr 17 2010

Keywords

Comments

Square grid, 4 neighbors per cell, turn ON iff exactly 1, 3 or 4 neighbors are ON; once ON, cells stay ON.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 928.

Links

Crossrefs

Cf. A139250, A151725, A147562, A147582, A169648, A169649, A169707, A169708, A169710.

Programs

  • Mathematica
    Map[Function[Apply[Plus,Flatten[ #1]]], CellularAutomaton[{ 1006, {2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},100]]

A350632 a(n) is the total number of ON cells at stage n of a cellular automaton where cells are turned ON when they have one or two neighbors ON (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 21, 45, 57, 85, 121, 177, 189, 217, 253, 329, 373, 465, 557, 721, 737, 765, 801, 877, 921, 1013, 1105, 1301, 1377, 1485, 1601, 1805, 1985, 2221, 2449, 2873, 2909, 2937, 2973, 3049, 3093, 3185, 3277, 3473, 3549, 3657, 3773, 3977, 4157, 4393, 4621, 5113
Offset: 0

Views

Author

Rémy Sigrist, Jan 08 2022

Keywords

Comments

On the infinite square grid, start with all cells OFF.
Turn a single cell to the ON state.
At each subsequent step, each cell with exactly one or two neighbors ON is turned ON, and everything that is already ON remains ON.
Here "neighbor" refers to the eight adjacent and diagonally adjacent cells in the Moore neighborhood.

Examples

			The first 5 generations can be depicted as follows:
         . . . . . . . . . . .
         . 5 5 . . . . . 5 5 .
         . 5 4 4 4 4 4 4 4 5 .
         . . 4 3 3 . 3 3 4 . .
         . . 4 3 2 2 2 3 4 . .
         . . 4 . 2 1 2 . 4 . .
         . . 4 3 2 2 2 3 4 . .
         . . 4 3 3 . 3 3 4 . .
         . 5 4 4 4 4 4 4 4 5 .
         . 5 5 . . . . . 5 5 .
         . . . . . . . . . . .
- so a(0) = 0,
     a(1) = 0 + 1 = 1,
     a(2) = 1 + 8 = 9,
     a(3) = 9 + 12 = 21,
     a(4) = 21 + 24 = 45,
     a(5) = 45 + 12 = 57.

Links

Crossrefs

Cf. A147562, A151725, A160414, A256530, A350633, A350639.

A151729 a(n) = (A151728(n)-1)/2.

Original entry on oeis.org

0, 2, 2, 5, 3, 7, 9, 11, 3, 7, 10, 14, 14, 24, 29, 23, 3, 7, 10, 14, 14, 24, 30, 26, 14, 25, 35, 43, 53, 78, 81, 47, 3, 7, 10, 14, 14, 24, 30, 26, 14, 25, 35, 43, 53, 78, 82, 50, 14, 25, 35, 43, 53, 79, 87, 67, 54, 86, 114, 140, 185, 238, 209, 95
Offset: 0

Views

Author

N. J. A. Sloane, Jun 14 2009

Keywords

Crossrefs

Cf. A151725-A151728.

A151727 If A151726 is written as a triangle, this is what the rows converge to.

Original entry on oeis.org

4, 20, 20, 44, 28, 60, 76, 92, 28, 60, 84, 116, 116, 196, 236, 188, 28, 60, 84, 116, 116, 196, 244, 212, 116, 204, 284, 348, 428, 628, 652, 380, 28, 60, 84, 116, 116, 196, 244, 212, 116, 204, 284, 348, 428, 628, 660, 404, 116, 204, 284, 348, 428
Offset: 0

Views

Author

N. J. A. Sloane, Jun 14 2009

Keywords

Crossrefs

Cf. A151725, A151726, A151729. Equals 4*A151728.

A249668 Population of the n-th generation of the pattern 101//010 in the Life Without Death cellular automaton.

Original entry on oeis.org

3, 4, 7, 10, 15, 20, 25, 30, 38, 47, 55, 63, 69, 79, 91, 96, 105, 112, 124, 134, 139, 144, 152, 157, 163, 168, 176, 183, 187, 195, 205, 217, 223, 229, 239, 247, 259, 273, 285, 289, 295, 303, 311, 323, 334, 339, 343, 351, 363, 375, 383, 389, 397, 405, 413, 423
Offset: 0

Views

Author

Eric M. Schmidt, Nov 03 2014

Keywords

Comments

Each generation, a cell turns on if it has exactly three neighbors that are on. Cells never turn off.
This pattern grows indefinitely. No other connected 3-celled pattern does so.

Examples

			Generation 0:
101
010
Generation 1:
111
010
Generation 2:
010
111
111
Generation 3:
111
111
111
010
Generation 4:
00100
01110
11111
01110
01110

Links

Crossrefs

Cf. A097981, A151725, A151731.

Formula

For n >= 2108, a(n+6) = a(n) + 260. - Eric M. Schmidt, Nov 04 2014
For n >= 2115, a(n) = a(n-1) + a(n-6) - a(n-7). - Eric M. Schmidt, Nov 05 2014
Previous Showing 11-15 of 15 results.

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

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