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 A385525 #32 Aug 06 2025 00:08:46 %S A385525 327080,84,52,32,18,24,24,24,24,24,18,24,24,24,24,24,24,24,24,30,24, %T A385525 30,30,24,30,30,30,30,30,30,30,30,30,30,30,30,30,36,30,36,36,30,36,36, %U A385525 36,36,36,36,36,36,36,36,36,36,36,36,36,40,36,36,40,36,40,42 %N A385525 Consider the graph whose vertices are the points of the n-dimensional cubic lattice with points connected by all integer-length diagonals that traverse all n dimensions and do not intersect intermediate points. a(n) is the total length of the shortest possible closed walk in this graph via noncongruent diagonals of the same length. %C A385525 Other than a(2) whose walk is comprised of 8 diagonal segments, all known terms are produced by 3- or 4-segment walks, including some with examples of both. It is conjectured that this holds true for all n >= 3. %C A385525 For n = 2, the walk segments are the hypotenuses of noncongruent primitive Pythagorean triangles. %C A385525 The offset is 2, because even though the graph could be defined in dimension 1 (the vertices would be the points of Z, with each point connected to its two neighbors), it would not contain any closed paths. %C A385525 Removing the constraint that the diagonal segments must all have the same length gives A386251. All such walks in 2 dimensions, by diagonal segment length rather than total walk length, gives A386671. %H A385525 Charles L. Hohn, <a href="/A385525/a385525.png">a(2), graphical view</a> %H A385525 Charles L. Hohn, <a href="/A385525/a385525.gif">a(3), graphical view, animated</a> %e A385525 a(2) = 327080 because [3636, 40723] + [8844, 39917] + [11603, -39204] + [38076, -14893] + [-37523, -16236] + [35844, -19667] + [-34387, -22116] + [-26093, 31476] = [0, 0] and 8 segments * length 40885 = 327080, which is the smallest example for n = 2. %e A385525 a(3) = 84: [16, 11, 8] + [-13, 4, 16] + [-8, -19, -4] + [5, 4, -20] = [0, 0, 0] and 4 * 21 = 84. %e A385525 a(4) = 52: [8, 8, 5, 4] + [-9, -6, 6, 4] + [-7, -4, -10, 2] + [8, 2, -1, -10] = [0, 0, 0, 0] and 4 * 13 = 52. %Y A385525 Cf. A386251, A386671. %Y A385525 Cf. A020882 (diagonals in 2 dimensions), A096910 (diagonals in 3 dimensions). %K A385525 nonn %O A385525 2,1 %A A385525 _Charles L. Hohn_, Jul 30 2025