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 A385176 #9 Jul 23 2025 16:13:35 %S A385176 1,-1,2,2,2,3,2,-1,3,4,-2,3,3,4,5,-2,3,-1,4,5,6,3,3,4,4,5,6,7,3,-2,4, %T A385176 -1,5,6,7,8,3,-2,4,5,5,6,7,8,9,-3,4,4,5,-1,6,7,8,9,10,-3,4,-2,5,6,6,7, %U A385176 8,9,10,11,-3,4,-2,5,6,-1,7,8,9,10,11,12,4,4,5,5,6,7,7,8,9,10,11,12,13 %N A385176 Positive half of inverse speed permutation array. Square array A(n,k), n >= 0, k >= 0, read by ascending antidiagonals. %C A385176 Particles labeled with nonzero integers j start at time t = 0 at x = 2k (offset from the origin) on a straight line. Each particle, j, moves at speed -1/j, so crosses the origin at time t = 2j^2. T(n,k) gives the label of the particle in the line segment (2k, 2k+2) at time t = 2n+1. %C A385176 It is easy to determine that particles labeled i and -j cross at x = 2*(i-j) at time t = 2ij, and that (for t > 0) a particle crosses x = 2k only when encountering a particle heading in the opposite direction. So at t = 2n+1 there is exactly one particle in each segment (2k, 2k+2) and the particle labels define a bi-infinite permution of the nonzero integers. For the terms of this sequence, we restrict k >= 0; and taking the absolute values of the terms in each row gives a permutation of the positive integers. Moreover, the differences between row n-1 and row n consist of exchanges of paired divisors of -n. %C A385176 The halved positions, k, at which particles encounter a segment boundary x = 2k at t = 2n are given by row n of A368312. So when that row starts with a 0, this indicates a particle crossing the origin. On the other hand, the nonzero terms, k, of row t of A211343 indicate the segment midpoints x = 2k-1 that are encountered by particles at time t, with terms in odd (respectively even) columns corresponding to positive-labeled (respectively negative-labeled) particles. %e A385176 Square array A(n,k) begins: %e A385176 n t\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 %e A385176 --------+----------------------------------------------------------------- %e A385176 0 1 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 %e A385176 1 3 | -1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 %e A385176 2 5 | 2, -1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 %e A385176 3 7 | 2, 3, -1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 %e A385176 4 9 | -2, 3, 4, -1, 5, 6, 7, 8, 9, 10, 11, 12, 13 %e A385176 5 11 | -2, 3, 4, 5, -1, 6, 7, 8, 9, 10, 11, 12, 13 %e A385176 6 13 | 3, -2, 4, 5, 6, -1, 7, 8, 9, 10, 11, 12, 13 %e A385176 7 15 | 3, -2, 4, 5, 6, 7, -1, 8, 9, 10, 11, 12, 13 %e A385176 8 17 | 3, 4, -2, 5, 6, 7, 8, -1, 9, 10, 11, 12, 13 %e A385176 9 19 | -3, 4, -2, 5, 6, 7, 8, 9, -1, 10, 11, 12, 13 %e A385176 10 21 | -3, 4, 5, -2, 6, 7, 8, 9, 10, -1, 11, 12, 13 %e A385176 11 23 | -3, 4, 5, -2, 6, 7, 8, 9, 10, 11, -1, 12, 13 %e A385176 12 25 | 4, -3, 5, 6, -2, 7, 8, 9, 10, 11, 12, -1, 13 %Y A385176 Cf. A211343, A368312. %K A385176 sign,tabl %O A385176 0,3 %A A385176 _Peter Munn_, Jun 20 2025