A343530 -id:A343530 - cp's OEIS frontend

A358048 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that every number shares a digit with each of its eight surrounding neighbors.

0, 10, 20, 30, 40, 50, 60, 70, 80, 18, 100, 12, 2, 23, 102, 90, 49, 104, 101, 103, 16, 105, 106, 107, 78, 8, 81, 1, 21, 22, 24, 25, 26, 29, 19, 9, 39, 91, 14, 11, 13, 15, 17, 31, 41, 51, 71, 87, 28, 38, 48, 108, 61, 112, 27, 32, 34, 42, 52, 62, 69, 59, 79, 89, 83, 93, 94, 109, 110, 111, 113

Offset: 0

Eric Angelini and Scott R. Shannon, Oct 27 2022

The sequence is conjectured to be a permutation of the nonnegative integers.

			The square spiral begins:
                           .
                           .
   49--90--102---23---2   22
   |                  |    |
  104  40---30---20   12  21
   |   |          |   |    |
  101  50    0---10  100   1
   |   |              |    |
  103  60---70---80---18  81
   |                       |
   16--105--106--107--78---8
.
a(11) = 12 as when the eleventh square is filled its surrounding neighbors are 10, 20, 100, and 12 is the smallest unused number that shares a digit with each of these three numbers.

Eric Angelini, More inspiring spirals (1), personal blog CinquanteSignes.blogspot.com, Oct. 23, 2022.
Gleb Ivanov, Python program.

Cf. A358021, A344325, A344367, A354111, A343530.

A344367 Squares visited on a spirally numbered board when stepping to the closest unvisited square that contains a number that shares one or more digits with the number of the current square. If two or more such squares are the same distance away the one with the smaller number is chosen.

Original entry on oeis.org

1, 11, 10, 12, 13, 14, 15, 16, 17, 18, 19, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 23, 22, 21, 20, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 2, 52, 51, 50, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 59, 58, 57, 56, 55, 54, 53, 125, 124, 123, 122, 121, 120

Offset: 1

Scott R. Shannon and Eric Angelini, May 16 2021

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12  29
   |   |       |   |   |
  19   6   1---2  11  28
   |   |           |   |
  20   7---8---9--10  27
   |                   |
  21--22--23--24--25--26
.
a(2) = 11. There are three squares 2 units away from the starting square 1 that also contain the digit 1 - 11, 15, and 19. Of these 11 is the smallest so is the square stepped to.
a(3) = 10. Of the two adjacent squares to 11 that also contain the digit 1 the square 10 is the smallest.
a(4) = 12. This is the only unvisited square within 2 units of a(3) = 10 that also contains the digit 1.
a(12) = 39. This is the only unvisited square within sqrt(2) units of a(11) = 19 that contains either the digit 3 or 9. It is also the first square stepped to that does not share the digit 1 with the previous square.

Scott R. Shannon, Image of the first 1000 steps.. The colors are graduated across the spectrum to show the relative step order. The lowest unvisited square, 4, is marked with a yellow dot.
Scott R. Shannon, Image of the first 200000 steps.

Cf. A343530, A296030, A174344, A274923, A067581, A330979, A332767.

A358021 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that no number shares a digit with any of its eight surrounding neighbors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 44, 10, 55, 11, 20, 13, 66, 12, 33, 14, 22, 15, 23, 16, 24, 57, 18, 25, 36, 27, 48, 26, 34, 56, 47, 28, 50, 49, 58, 60, 29, 35, 67, 38, 40, 77, 59, 37, 19, 68, 39, 46, 70, 41, 80, 31, 65, 90, 17, 88, 21, 93, 51, 43, 69, 71, 32, 111, 73, 81, 64, 72, 89, 42, 91

Offset: 0

Scott R. Shannon and Eric Angelini, Oct 24 2022

The sequence is finite; after a(264) = 177 the square at coordinate (-8,-1) relative to the starting square is reached which has the four neighbors 177, 409, 256, 308. These numbers include all digits 0 to 9 so the next term does not exist. The largest used number is a(241) = 7777 while smallest unused number is 120.

			The square spiral begins:
                       .
                       .
  13--20--11--55--10  36
   |               |   |
  66   4---3---2  44  25
   |   |       |   |   |
  12   5   0---1  30  18
   |   |           |   |
  33   6---7---8---9  57
   |                   |
  14--22--15--23--16--24
.
a(10) = 30 as when the tenth square is filled its surrounding neighbors are 1, 2, 8, 9, and 30 is the smallest unused number that does not contain those digits.

Cf. A358048, A344325, A344367, A354111, A343530.

A358254 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that the sum of the eight numbers around any chosen number ends in the chosen number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 12, 8, 9, 10, 11, 13, 15, 23, 14, 16, 18, 21, 17, 19, 29, 25, 33, 20, 22, 26, 28, 120, 24, 27, 87, 58, 125, 88, 30, 31, 97, 124, 45, 187, 32, 34, 73, 132, 55, 49, 42, 35, 36, 95, 195, 59, 98, 863, 37, 38, 130, 104, 129, 62, 736, 67, 39, 40, 115, 131, 48, 748, 82, 208, 41

Offset: 0

Eric Angelini and Scott R. Shannon, Nov 05 2022

The first two numbers placed when a new row or column is formed around the spiral do not complete a 3x3 block of numbers, thus they can be the lowest two numbers that have not appeared. Therefore the sequence a permutation of the nonnegative integers.

In the first 10000 terms the largest value is a(8526) = 9874588 while the smallest unused number is 902.

			The square spiral begins:
                       .
                       .
  14--23--15--13--11  120
   |               |   |
  16   4---3---2  10  28
   |   |       |   |   |
  18   5   0---1   9  26
   |   |           |   |
  21   6---7--12---8  22
   |                   |
  17--19--29--25--33--20
.
a(8) = 12 as when the ninth cell is filled it completes a ring of eight numbers around the central cell with number 0, therefore the sum of these eight numbers must end in 0. The sum around the central cell when the eighth cell is filled is 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28, and the lowest unused number that can be added so that the sum ends in 0 is 28 + 12 = 40, so a(8) = 12.
a(29) = 120 as when the thirtieth cell is filled the sum of the previous numbers around the number 10 is 13 + 11 + 2 + 28 + 1 + 9 + 26 = 90, and since 20 has already appeared the smallest unused number that can be added to 90 to form a number that ends in 10 is 120.

Scott R. Shannon, Table of n, a(n) for n = 0..5000
Eric Angelini, 9 frames and 1 super-frame, personal blog CinquanteSignes.blogspot.com, Oct. 30, 2022.

Cf. A358048, A358021, A358021, A344325, A344367, A354111, A343530.

A358129 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that no number shares a digit with any of its four orthogonally adjacent neighbors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 11, 22, 10, 24, 13, 25, 16, 23, 14, 26, 15, 28, 17, 30, 12, 33, 18, 27, 19, 32, 40, 29, 31, 42, 36, 21, 34, 50, 41, 35, 44, 37, 45, 38, 46, 39, 47, 53, 48, 55, 49, 52, 43, 56, 70, 51, 60, 57, 61, 54, 63, 58, 64, 59, 66, 71, 62, 73, 68, 72, 65, 74, 69, 77, 90, 75

Offset: 0

Scott R. Shannon and Eric Angelini, Oct 30 2022

This sequence is similar to A358021 except here only the four orthogonally adjacent neighbors are considered. It is unknown if the sequence eventually terminates; if it does, it does so after more than 1.5 million terms. In that range on numerous occasions the only available number is one containing only the digit 7. For example a(862833) = 7...7, where there are forty-nine 7's.

			The square spiral begins:
                       .
                       .
  25--13--24--10--22  19
   |               |   |
  16   4---3---2  11  27
   |   |       |   |   |
  23   5   0---1  20  18
   |   |           |   |
  14   6---7---8---9  33
   |                   |
  26--15--28--17--30--12
.
a(10) = 20 as when the tenth cell is filled its two orthogonal neighbors use the digits 1 and 9, and 20 is the smallest unused number that does not contain either of those digits.

Scott R. Shannon, Image of the first 500000 terms. The values are scaled across the spectrum from red to violet, with the value ranges increasing towards the violet end to give more color weighting to the larger numbers.

Cf. A358021, A358048, A344325, A344367, A354111, A343530.

cp's OEIS Frontend

A358048 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that every number shares a digit with each of its eight surrounding neighbors.

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A344367 Squares visited on a spirally numbered board when stepping to the closest unvisited square that contains a number that shares one or more digits with the number of the current square. If two or more such squares are the same distance away the one with the smaller number is chosen.

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A358021 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that no number shares a digit with any of its eight surrounding neighbors.

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A358254 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that the sum of the eight numbers around any chosen number ends in the chosen number.

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A358129 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that no number shares a digit with any of its four orthogonally adjacent neighbors.

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