cp's OEIS Frontend

We define earliest as the number with the smallest magnitude, and where a positive number is considered earlier than its negative value.

To ensure the sequence is infinite, other than tracking the numbers that have already appeared in the spiral, one must also calculate the numbers around the outside of the spiral whose values have been fixed by their nearest neighbor on the spiral already having three neighbors with set values. These numbers may have similar neighbors which will force the calculation of other values in a cascade of new fixed numbers that appear as a square diamond shape around the currently filled spiral. When the lowest available number is to be chosen one must ensure it does not equal any of these predetermined values. One additional check must be performed to ensure the sequence is infinite; see the examples below. Unlike A358151 here the numbers along the rows/columns are smaller towards the ends of the rows/columns and larger in the middle. See the linked images.

The values that are positive and negative make a pattern around the spiral axes; see the second linked image. There is no occurrence in the first 500000 terms where a positive number is surrounded by eight negative numbers, or vice versa. It is unknown is this can occur.

The sequence is conjectured to be a permutation of the integers. The values grow rapidly in size, e.g., in the first 1000 terms the largest magnitude number is a(976) = -2492394, while the largest magnitude number in the first 500000 terms is a(499495) = -749...021, a number containing 147 digits.

The numbers are determined by a recursive backtracking search starting from a(1) = 1. Additional optimization of available candidate values is achieved by checking the undetermined squares on the spiral that will form part of the 3 x 3 block of numbers the term currently being determined lies in; these are checked for numbers that they cannot contain and comparing those with similar values in other undetermined squares in the same 3 by 3 block. If all of these squares contain the same excluded value or values then these form the list of candidate numbers the current square must contain. If no such excluded shared value exists then the current square's list of candidate values is all those numbers that are not excluded by its neighbors in any surrounded 3 by 3 block of numbers.

Although significant backtracking is required using the above algorithm one finds that the resulting numbers form a repeating pattern of three values in three rows and columns in each of the four orthogonal quadrants around the starting square; this implies the sequence is indeed infinite in the resulting pattern of rows and columns. See the attached colored image. One finds, assuming a counter-clockwise numbered spiral, that the only repeating rows or columns that cross the quadrant boundaries are the column of values 5,6,7,5,6,7,5... on the left side of the spiral relative to the starting square, and the column 5,7,6,5,7,6,5,... on the right side.

As the terms are not distinct the first two numbers of any new row or column will always be zero. In the first 500000 terms the last zero that is not at the beginning of a row or column is a(190) = 0. Is it unknown if more such zeros exist. In the same range the smallest positive numbers not yet occurring are 5, 9, 11, 12, 15, 19, 20, ... . It is unknown if all integers eventually appear. The terms increase rapidly in size; in the first 500000 terms the largest positive term is a(499848) = 1267...5398, a number with 226 digits.