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 A343031 #25 Apr 05 2021 03:55:50 %S A343031 1,2,3,3,7,9,4,11,12,20,5,15,24,24,39,6,19,33,34,42,67,7,23,42,58,54, %T A343031 71,107,8,27,54,74,75,81,110,160,9,31,66,90,115,105,122,164,229,10,35, %U A343031 75,110,140,141,143,174,232,315,11,39,84,130,165,201,183,198,244,319,421 %N A343031 Table read by antidiagonals: T(n, k) is the sum of the numbers on the k-th line of length n when these lines are drawn on the square spiral, where each line contains numbers which sum to the minimum possible value, and each number on the spiral can only be in one line. %C A343031 Lines of length zero (a single point) and one (two points) can cover the entire square spiral without missing any numbers. %C A343031 For lines with even numbered length the pattern of lines is very regular, with all lines along the spiral lines of the square spiral, and regular triangles of uncovered numbers along the four diagonals of the spiral. See the linked images with even n. %C A343031 For odd length lines the pattern formed is more random, with some quadrants have regions, or the entire quadrant, with lines that are orthogonal to the spiral lines, and the triangles of uncovered values becomes more random along the spiral diagonals. See the linked images with odd n. %C A343031 For n>=2 the smallest spiral number that is not covered by any line is n^2+4n+4. %H A343031 Scott R. Shannon, <a href="/A343031/a343031.png">Image for n=1, k=1..4000</a>. The image can be zoomed in to see the numbers of the square spiral. In this and other images the colors are graduated around the spectrum to show the lines relative placement order. %H A343031 Scott R. Shannon, <a href="/A343031/a343031_1.png">Image for n=2, k=1..2000</a>. %H A343031 Scott R. Shannon, <a href="/A343031/a343031_2.png">Image for n=3, k=1..2000</a>. %H A343031 Scott R. Shannon, <a href="/A343031/a343031_3.png">Image for n=4, k=1..2000</a>. %H A343031 Scott R. Shannon, <a href="/A343031/a343031_5.png">Image for n=9, k=1..2000</a>. %H A343031 Scott R. Shannon, <a href="/A343031/a343031_4.png">Image for n=10, k=1..2000</a>. %H A343031 Scott R. Shannon, <a href="/A343031/a343031_6.png">Image for n=17, k=1..2000</a>. %H A343031 Scott R. Shannon, <a href="/A343031/a343031_8.png">Image for n=20, k=1..2000</a>. %H A343031 Scott R. Shannon, <a href="/A343031/a343031_7.png">Image for n=21, k=1..2000</a>. %F A343031 T(0,k) = k. %F A343031 T(1,k) = 3 + 4(k-1). %e A343031 The square spiral used is: %e A343031 . %e A343031 17--16--15--14--13 . %e A343031 | | . %e A343031 18 5---4---3 12 29 %e A343031 | | | | | %e A343031 19 6 1---2 11 28 %e A343031 | | | | %e A343031 20 7---8---9--10 27 %e A343031 | | %e A343031 21--22--23--24--25--26 %e A343031 . %e A343031 The table begins: %e A343031 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... %e A343031 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, ... %e A343031 9, 12, 24, 33, 42, 54, 66, 75, 84, 96, 105, 114, ... %e A343031 20, 24, 34, 58, 74, 90, 110, 130, 154, 178, 194, 210, ... %e A343031 39, 42, 54, 75, 115, 140, 165, 195, 225, 260, 295, 335, ... %e A343031 67, 71, 81, 105, 141, 201, 237, 273, 315, 357, 405, 453, ... %e A343031 107, 110, 122, 143, 183, 238, 322, 371, 420, 476, 532, 595, ... %e A343031 160, 164, 174, 198, 234, 294, 372, 484, 548, 612, 684, 756, ... %e A343031 229, 232, 244, 265, 305, 360, 444, 549, 693, 774, 855, 945, ... %e A343031 315, 319, 329, 353, 389, 449, 527, 639, 775, 955, 1055, 1155, ... %e A343031 421, 424, 436, 457, 497, 552, 636, 741, 885, 1056, 1276, 1397, ... %e A343031 548, 552, 562, 586, 622, 682, 760, 872, 1008, 1188, 1398, 1662, ... %Y A343031 Cf. A340974, A341160, A341363, A341278, A341327, A174344, A274923, A296030, A275161. %K A343031 nonn,tabl %O A343031 0,2 %A A343031 _Scott R. Shannon_, Apr 03 2021