This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A353588 #17 Sep 11 2022 10:22:28 %S A353588 0,1,2,6,13,17,3,16,32,4,63,375,20,1628,45,36,43,25189,138507681, %T A353588 3727222924,5,123772,1299458 %N A353588 Simultaneously fill a square spiral and an infinite square array by antidiagonals with distinct nonnegative integers so that the sum of numbers in any 2 X 2 square equals a square. %C A353588 Each term is the smallest possible solution satisfying the constraints and such that the sequence can be extended thereafter. %C A353588 We have no formal proof that the sequence is indeed well defined, i.e., infinite. However, at points which are at the corners of the square spiral or on the first row or column of the square array, one always has infinitely many solutions. It appears that this is sufficient to infinitely extend the sequence without the need of going back further than the last one of these points. %C A353588 Next term a(23) >= 371885618. %e A353588 The first 22 terms form the following square spiral, going clockwise after starting at the center towards the left: %e A353588 5 ----> 123772 --> 1299458 ---> ... %e A353588 3727222924 3 ------> 16 -----> 32 ------> 4 %e A353588 138507681 17 start: 0 ------> 1 63 %e A353588 25189 13 <------- 6 <------ 2 375 %e A353588 43 <----- 36 <------- 45 <---- 1628 <---- 20 %e A353588 The square array filled with the same terms starts: %e A353588 0 1 6 3 63 36 123772 %e A353588 2 13 16 375 43 1299458 %e A353588 17 32 20 25189 ... %e A353588 4 1628 138507681 ... %e A353588 45 3727222924 ... %e A353588 5 ... %e A353588 ... %e A353588 After the initial terms (0, 1, 2), there is a single constraint on a(3), which comes from the square spiral, where 0 + 1 + 2 + a(3) must sum to a square. Since the number 1 was already used, a(3) = 6 is the smallest possible choice. %e A353588 The "inner" elements of the antidiagonals, like a(4) = 13, must satisfy the constraint arising from their left, top and top-left neighbors, with which they have to sum to a square. This becomes nontrivial if they also have, at the same time, three already fixed neighbors on the square spiral. This happens when the term is not the first or last one on a straight line on the square spiral. %e A353588 For example the term a(7) = 16 must at the same time satisfy 0 + 3 + 17 = x^2 (from the spiral) and 1 + 6 + 13 + a(3) = y^2 (from the array). In this case, the sum of the neighbors is the same, so there is actually only one constraint to satisfy. The smallest solution would be 5, but it turns out that this does not allow to find a solution for the next terms: the neighbors would sum to 6 and 32, respectively, which cannot be completed with the same term to two distinct squares. This can be proved by checking all possible squares s^2 up to s = 13: At this point, (s+1)^2 - s^2 = 2s+1 becomes larger than the difference of the two sums, 32 - 6 = 26. Thus, the solution 5 must be excluded at index n = 7. %e A353588 Similarly, one must exclude solutions 12, 27 and 44 at index n = 10, and solution 1669 at index n = 21. %e A353588 The term a(8) must satisfy 0 + 1 + 16 + a(8) = x^2 from the square spiral, and 2 + 13 + 17 + a(8) = y^2 on the antidiagonal of the square array. The smallest solution is a(8) = 32. %Y A353588 Cf. A000290 (squares); A174344, A274923 (x & y coordinate of the square spiral's n-th point); A002262, A025581 (row & column number of the table's n-th entry when read by falling antidiagonals). %Y A353588 Cf. A337115 (square spiral such that any 2 X 2 square sums to a square). %K A353588 nonn,more %O A353588 0,3 %A A353588 _Eric Angelini_ and _M. F. Hasler_, May 29 2022