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

1,2

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.

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

Example

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

Prog

(Python) # see linked program

Crossrefs

Cf. A090915, A217010, A337822.

Sequence in context: A340711 A242304 A227415 * A309156 A298042 A051430

Adjacent sequences: A361204 A361205 A361206 * A361208 A361209 A361210

Keyword

nonn,easy

Author

John Tyler Rascoe, Mar 04 2023

Status

approved