A361207 An infinite 2d grid is filled with the positive integers by placing them clockwise around the lowest number with open neighbors. a(n) is then the n-th term when the grid is read as a clockwise square spiral.

1, 2, 7, 3, 10, 4, 12, 5, 8, 16, 6, 15, 29, 17, 9, 20, 35, 21, 11, 23, 39, 24, 13, 18, 30, 46, 28, 14, 27, 45, 67, 47, 31, 19, 34, 53, 76, 54, 36, 22, 38, 58, 82, 59, 40, 25, 32, 48, 68, 92, 66, 44, 26, 43, 65, 91, 121, 93, 69, 49, 33, 52, 75, 102, 133, 103, 77

Offset: 1

Author

John Tyler Rascoe, Mar 04 2023

Keywords

• nonn
• easy

Comments

To begin, 1 is placed at square (x,y) = (0,0); this then becomes square s = 1. Integers are added sequentially to the open squares of the neighborhood around square s. The neighborhood of a square is defined as: east (x+1,y), south (x,y-1), west (x-1,y), and north (x,y+1). The order in which numbers may be added to a neighborhood is always east, south, west, then north.

Numbers are added to open squares in the neighborhood of square s following the given order. The next number added to the grid is always the smallest positive integer not yet present on the grid. If a filled square is encountered within the current square's neighborhood, the process moves to the next direction in the order. Once the process has cycled through all directions of the neighborhood of a given square s, the process is repeated at square s+1.

The filled grid is then read as a clockwise square spiral, oriented east starting at (0,0). a(n) is the n-th term along the square spiral.

Since each positive integer is added to the grid once, reading the grid as a spiral gives a permutation of the positive integers. Similar permutations can be created by expanding the neighborhood of s.

Examples

			The spiral begins:
.
                    41
.
                40  25  32
.
            39--24--13--18--30
.            |
        38  23  12---5-- 8--16  28
.            |   |           |
    37  22  11   4   1---2   6  14  26
.            |   |       |   |
        36  21  10---3---7  15  27
.            |               |
            35--20---9--17--29
.
                34  19  31
.
                    33
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..10000
• John Tyler Rascoe, Python program
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A090915, A217010, A337822.

Programs

Python

# see linked program