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
Keywords
Examples
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.
Links
- 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.
Comments