A126803 -id:A126803 - cp's OEIS frontend

A337405 A fractal spiral on a 2D square lattice, one digit per cell, starting at the origin with 0. The odd digits reproduce the spiral itself at another scale (design of the digits is shown below).

0, 1, 3, 2, 5, 7, 9, 4, 10, 6, 8, 21, 11, 20, 22, 23, 13, 24, 26, 28, 40, 42, 44, 25, 15, 17, 46, 48, 60, 62, 19, 31, 12, 33, 27, 35, 29, 14, 64, 66, 16, 37, 68, 41, 80, 18, 82, 84, 86, 43, 30, 88, 32, 200, 45, 34, 202, 47, 49, 201, 204, 36, 61, 206, 208, 220, 222, 63, 39, 65, 51, 67, 53, 224, 226

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 26 2020

Here's the font that's used; digits are separated horizontally by a single empty column and vertically by a single empty row. The same digit design was selected 52 years ago by Jonathan Vos Post.
.
%%%...%...%%%...%%%...%.%...%%%...%%%...%%%...%%%...%%%
%.%...%.....%.....%...%.%...%.....%.......%...%.%...%.%
%.%...%...%%%...%%%...%%%...%%%...%%%.....%...%%%...%%%
%.%...%...%.......%.....%.....%...%.%.....%...%.%.....%
%%%...%...%%%...%%%.....%...%%%...%%%.....%...%%%...%%%
.
The spiral's elements are the successive digits of the sequence. The sequence is the lexicographically earliest one of distinct nonnegative terms that starts exactly in the center of the zero formed by the first 12 odd digits (see below).

			The start of the spiral, with the odd digits forming a zero:
.
       2——3——1——3——2——4
       |              |
       2  9——4——1——0  .
       |  |        |  |
       2  7  0——1  6  .
       |  |     |  |  |
       0  5——2——3  8
       |           |
       2——1——1——1——2
.
The first eight turns of the spiral (the odd digits have brackets which should help the visualization of the scaled new digits):
.
[1]—[3]——2——[1]—[5]—[1]——0——[3]——2——[3]——2——2——[3]——4——2——[1]—[7]—[1]
.|                                                                 |
.2  [3]——2——[1]——2——[5]——6——[5]——8——[7]——0——2——[1]——4——2——[1]——6  [1]
.|   |                                                         |   |
[1] [9]  6——[9]—[5]—[5]——8——[1]—[5]—[7]——2——0——[3]——2——0——[5]  2  [1]
.|   |   |                                                 |   |   |
.0   4   0   2———0——[1]——2———0———4——[3]——6——6——[1]——2——0  [5]  0  [1]
.|   |   |   |                                         |   |   |   |
[1] [5]  4  [9] [3]—[7]——6———8———4——[1]——8——0——[1]——8  6  [9] [7] [1]
.|   |   |   |   |                                  |  |   |   |   |
.8   2   2   4   6   4———6———4———8———6———0——6———2   8  2   8   2   2
.|   |   |   |   |   |                          |   |  |   |   |   |
[9] [5]  8  [7] [1] [7]  2——[3]—[1]—[3]——2——4  [1]  2  0  [3] [1] [1]
.|   |   |   |   |   |   |                  |   |   |  |   |   |   |
.6  [1]  2   4   6  [1]  2  [9]——4——[1]——0  2  [9]  8  8  [7]  8   0
.|   |   |   |   |   |   |   |           |  |   |   |  |   |   |   |
[9] [9]  2   2   6  [5]  2  [7]  0——[1]  6  6  [3]  4  2  [1] [7] [5]
.|   |   |   |   |   |   |   |       |   |  |   |   |  |   |   |   |
.0   0   6   0   4  [1]  0  [5]——2——[3]  8  2  [1]  8  2  [7]  2  [1]
.|   |   |   |   |   |   |               |  |   |   |  |   |   |   |
[3] [1]  2   2   6  [5]  2——[1]—[1]—[1]——2  8  [1]  6  0  [3] [7] [1]
.|   |   |   |   |   |                      |   |   |  |   |   |   |
.2   2   2   4   4   2———4———4———2———4———0——4   2   4  2   8   4   4
.|   |   |   |   |                              |   |  |   |   |   |
[9] [9]  4  [3] [1]—[9]——2——[5]—[3]—[7]——2—[3]—[3] [3] 2  [5] [7] [1]
.|   |   |   |                                      |  |   |   |   |
.0  [7]  2  [5]——4———0———0———2———2——[3]——8——8———0——[3] 2  [7]  6  [1]
.|   |   |                                             |   |   |   |
.2  [7]  2——[3]—[5]—[7]——6——[1]—[5]—[5]——6—[9]—[3]—[3]—6  [5] [7] [3]
.|   |                                                     |   |   |
.0  [7]——4———4———2——[9]——8——[7]——8———2———4——2———0——[5]—8——[3]  8   .
.|                                                             |
[1]—[1]——4——[9]—[9]—[9]——2——[9]—[7]—[9]——0—[9]—[1]—[1]—2——[5]—[9]  .
.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..1628
Eric Angelini, Une suite pour la Vie
Eric Angelini, Une suite pour la Vie [Cached, with permission]
Tangente, Une spirale de chiffres, Hors-série Kiosque 76 (Nov 2020).

Cf. A126803 (design of the digits), A337115.

A370776 The maximum number of alive cells reached in Conway's Game of Life when starting with the first n primes in Ulam's spiral; or -1 if no such maximum exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 65, 56, 120, 56, 28, 133, 30, 160, 46, 24, 24, 25, 28, 30, 31, 31, 32, 32, 32, 35, 74, 39, 38, 38, 38, 39, 40, 42, 319, 319, 319, 319, 319, 46, 129, 93, 50, 50, 72, 72, 72, 72, 72, 72, 53, 53, 56, 56, 851, 851, 167, 167, 167, 167, 391

Offset: 1

Thomas Strohmann, Mar 01 2024

nonn

The initial alive cells are at coordinates x=A214664(i), y=A214665(i) for i=1..n.
For the first 7 terms of this sequence we have a(n)=n since those initial configurations do not lead to complex enough patterns that increase the number of alive cells beyond the initial number of alive cells.
The definition includes the possibility that a glider gun (or a similar pattern) is created which will result in an unbounded number of alive cells.

			n=1 to n=4 die out very quickly (within 3 steps). The maximum number of alive cells is simply the number of alive cells in the initial pattern, i.e., n.
n=5 is the first term that leads to somewhat interesting steps in the game of life simulation (although the maximum number of alive cells still does not exceed the initial number 5):
  . . . . . | . . . . . | . . . o . | . . . o . | . . . o . | . . . . .
  o . o . . | . o o o . | . . o . o | . . o . o | . . . o . | . . . . .
  . . o o . | . . o o . | . . o . o | . . . . . | . . . . . | . . . . .
  o . . . . | . . . . . | . . . . . | . . . . . | . . . . . | . . . . .
n=8 leads to a maximum number of 65 alive cells and stabilizes after 107 steps. Initial pattern:
  o . . . o |
  . o . o . |
  o . . o o |
  . o . . . |
n=15 reaches a maximum of 160 alive cells and is the first pattern that leads to having a glider (escaping in the northeast direction). Besides the glider, the stabilized pattern contains 4 blinkers, 3 blocks, 2 beehives and 1 ship.

Wikipedia, Ulam spiral.
Wikipedia, Conway's Game of Life.

Cf. A214664, A214665.

Cf. A152389, A126803, A099733, A179412.

cp's OEIS Frontend

A337405 A fractal spiral on a 2D square lattice, one digit per cell, starting at the origin with 0. The odd digits reproduce the spiral itself at another scale (design of the digits is shown below).

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