A355270 Lexicographically earliest sequence of positive integers on a square spiral such that the sum of adjacent pairs of numbers within each row, column and diagonal is distinct in that row, column and diagonal.

1, 1, 1, 1, 2, 2, 3, 2, 4, 3, 3, 4, 4, 3, 5, 4, 2, 4, 3, 5, 4, 4, 2, 3, 6, 4, 6, 5, 7, 6, 2, 6, 3, 2, 5, 8, 4, 3, 6, 6, 7, 3, 5, 7, 6, 8, 8, 7, 1, 2, 7, 5, 1, 2, 5, 8, 6, 4, 8, 5, 6, 9, 7, 1, 4, 10, 1, 1, 6, 3, 9, 12, 5, 1, 7, 2, 1, 6, 4, 1, 13, 6, 4, 7, 9, 12, 10, 7, 11, 1, 5, 2, 10, 7, 4, 5, 8

Offset

1,5

Comments

In the first 2 million terms the largest number is 1959, while the number 1, the most commonly occurring number, appears 10893 times. See the linked images.

Links

Table of n, a(n) for n=1..97.

Scott R. Shannon, Image of the first 2 million 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.

Scott R. Shannon, Distribution of a(n) for the first 2 million terms. The number 1 appears 10893 times. The second maximum occurs at n ~ 500.

Example

The spiral begins:
.
                                .
    4---8---5---2---3---6---2   :
    |                       |   :
    3   2---4---5---3---4   6   5
    |   |               |   |   |
    6   4   2---1---1   4   7   2
    |   |   |       |   |   |   |
    6   3   2   1---1   3   5   1
    |   |   |           |   |   |
    7   5   3---2---4---3   6   5
    |   |                   |   |
    3   4---4---2---3---6---4   7
    |                           |
    5---7---6---8---8---7---1---2
.
a(25) = 6 as when a(25) is placed, at coordinate (2,-2) relative to the starting square, its adjacent squares are a(10) = 3, a(9) = 4, a(24) = 3. The sums of adjacent pairs of numbers in a(25)'s column are 3 + 3 = 6, 3 + 4 = 7, 4 + 4 = 8, in its northwest diagonal are 4 + 1 = 5, 1 + 2 = 3, 2 + 2 = 4, and in its row are 3 + 2 = 5, 2 + 4 = 6, 4 + 4 = 8. Setting a(25) to 1 would create a sum of 5 with its diagonal neighbor 4, but 5 has already occurred as a sum on this diagonal. Similarly numbers 2, 3, 4 and 5 can be eliminated as they create sums with the three adjacent numbers, 3, 4, and 3, which have already occurred along the corresponding column, diagonal or row. This leaves 6 as the smallest number which creates new sums, namely 9, 10 and 9, with its three neighbors that have not already occurred along the corresponding column, diagonal and row.

Crossrefs

Cf. A355271, A274640, A275609, A307834.

Sequence in context: A366311 A355271 A205000 * A066241 A334857 A331614

Adjacent sequences: A355267 A355268 A355269 * A355271 A355272 A355273

Keyword

nonn

Author

Scott R. Shannon, Jun 26 2022

Status

approved