A355314 Lexicographically earliest sequence of positive integers on a square spiral such that the difference between all orthogonally adjacent pairs of numbers is distinct.

0, 0, 1, 3, 7, 12, 1, 7, 15, 1, 10, 23, 0, 17, 35, 54, 0, 27, 48, 72, 0, 26, 55, 83, 31, 0, 34, 69, 106, 39, 1, 41, 83, 126, 1, 45, 91, 140, 77, 128, 2, 57, 1, 61, 119, 183, 1, 93, 158, 1, 74, 143, 218, 0, 115, 192, 0, 79, 160, 244, 2, 87, 174, 1, 89, 185, 1, 166, 6, 101, 198, 296, 0, 101, 203, 1

Offset: 0

Author

Scott R. Shannon, Jun 28 2022

Keywords

• nonn

Comments

For larger n the sequences typically consists of a repeating pattern of three values - the first one is small, less than 5, a second larger value, and then a third even larger value, typically around double the previous value. However this pattern is occasionally broken by a fourth or fifth larger value which shifts the position of the subsequent repeating block of three values. This leads to the overall spiral pattern showing a uniform pattern of numbers crossed by random zig-zag lines of values not following the three-value pattern. See the linked color image.

Examples

			The spiral begins:
.
                                .
   91--45---1--126-83--41---1   :
    |                       |   :
   140  0--54--35--17---0  39  115
    |   |               |   |   |
   77  27   7---3---1  23  106  0
    |   |   |       |   |   |   |
   128 48  12   0---0  10  69  218
    |   |   |           |   |   |
    2  72   1---7--15---1  34  143
    |   |                   |   |
   57   0--26--55--83--31---0  74
    |                           |
    1--61--119-183--1--93--158--1
.
.
a(8) = 15 as when a(8) is placed, at coordinate (1,-1) relative to the starting square, its two orthogonally adjacent squares are a(1) = 0 and a(7) = 7. The ten previously occurring differences between all orthogonally adjacent pairs up to a(7) are 0, 1, 2, 3, 4, 5, 6, 7, 11, 12. The lowest unused difference is 8 thus a(8) = 15 can be chosen as it results in differences with its two orthogonal neighbors of 15 - 7 = 8 and 15 = 0 = 15, neither of which has previously occurred.
		

Links

• Scott R. Shannon, Image of the first 500000 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.

Crossrefs

Cf. A307834, A275609, A274640, A355270.