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.

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

Author

Eric Angelini and Scott R. Shannon, Nov 05 2022

Keywords

• nonn
• base

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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