A355314 - OEIS

A355314

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

%I #11 Aug 09 2022 06:47:27

%S 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,

%T 69,106,39,1,41,83,126,1,45,91,140,77,128,2,57,1,61,119,183,1,93,158,

%U 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

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

%C 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.

%H Scott R. Shannon, <a href="/A355314/a355314.png">Image of the first 500000 terms</a>. 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.

%e The spiral begins:

%e .

%e 91--45---1--126-83--41---1 :

%e | | :

%e 140 0--54--35--17---0 39 115

%e | | | | |

%e 77 27 7---3---1 23 106 0

%e | | | | | | |

%e 128 48 12 0---0 10 69 218

%e | | | | | |

%e 2 72 1---7--15---1 34 143

%e | | | |

%e 57 0--26--55--83--31---0 74

%e | |

%e 1--61--119-183--1--93--158--1

%e .

%e 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.

%Y Cf. A307834, A275609, A274640, A355270.

%K nonn

%O 0,4

%A _Scott R. Shannon_, Jun 28 2022