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 A360853 #10 Feb 16 2025 08:34:04 %S A360853 0,0,0,1,1,1,4,5,5,4,10,14,21,14,10,20,30,58,58,30,20,35,55,125,236, %T A360853 125,55,35,56,91,231,720,720,231,91,56,84,140,385,1754,4040,1754,385, %U A360853 140,84,120,204,596,3654,15550,15550,3654,596,204,120 %N A360853 Array read by antidiagonals: T(m,n) is the number of induced cycles in the rook graph K_m X K_n. %C A360853 Induced cycles are sometimes called chordless cycles (but some definitions require chordless cycles to have a cycle length of at least 4). See A360849 for the version that excludes triangles. %H A360853 Andrew Howroyd, <a href="/A360853/b360853.txt">Table of n, a(n) for n = 1..1275</a> %H A360853 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>. %H A360853 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cycle_(graph_theory)">Cycle (graph theory)</a>. %F A360853 T(m,n) = A360849(m,n) + A360855(m,n). %F A360853 T(m,n) = T(n,m). %e A360853 Array begins: %e A360853 ========================================================== %e A360853 m\n| 1 2 3 4 5 6 7 8 ... %e A360853 ---+------------------------------------------------------ %e A360853 1 | 0 0 1 4 10 20 35 56 ... %e A360853 2 | 0 1 5 14 30 55 91 140 ... %e A360853 3 | 1 5 21 58 125 231 385 596 ... %e A360853 4 | 4 14 58 236 720 1754 3654 6808 ... %e A360853 5 | 10 30 125 720 4040 15550 45395 109840 ... %e A360853 6 | 20 55 231 1754 15550 114105 526505 1776676 ... %e A360853 7 | 35 91 385 3654 45395 526505 4662721 24865260 ... %e A360853 8 | 56 140 596 6808 109840 1776676 24865260 256485936 ... %e A360853 ... %o A360853 (PARI) T(m, n) = m*binomial(n,3) + n*binomial(m,3) + sum(j=2, min(m, n), binomial(m, j)*binomial(n, j)*j!*(j-1)!/2) %Y A360853 Main diagonal is A360854. %Y A360853 Rows 1..2 are A000292(n-2), A000330(n-1). %Y A360853 Cf. A360196, A360849, A360851 (induced paths), A360855 (triangles). %K A360853 nonn,tabl %O A360853 1,7 %A A360853 _Andrew Howroyd_, Feb 24 2023