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 A339190 #22 Feb 16 2025 08:34:01 %S A339190 3,4,4,8,16,8,16,120,120,16,32,744,2830,744,32,64,4922,50354,50354, %T A339190 4922,64,128,31904,1003218,2462064,1003218,31904,128,256,208118, %U A339190 19380610,139472532,139472532,19380610,208118,256,512,1354872,378005474,7621612496,22853860116,7621612496,378005474,1354872,512 %N A339190 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of (undirected) Hamiltonian cycles on the n X k king graph. %H A339190 Seiichi Manyama, <a href="/A339190/b339190.txt">Antidiagonals n = 2..12, flattened</a> %H A339190 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a> %H A339190 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a> %H A339190 <a href="/index/Gra#graphs">Index entries for sequences related to graphs, Hamiltonian</a> %F A339190 T(n,k) = T(k,n). %e A339190 Square array T(n,k) begins: %e A339190 3, 4, 8, 16, 32, 64, ... %e A339190 4, 16, 120, 744, 4922, 31904, ... %e A339190 8, 120, 2830, 50354, 1003218, 19380610, ... %e A339190 16, 744, 50354, 2462064, 139472532, 7621612496, ... %e A339190 32, 4922, 1003218, 139472532, 22853860116, 3601249330324, ... %e A339190 64, 31904, 19380610, 7621612496, 3601249330324, 1622043117414624, ... %o A339190 (Python) %o A339190 # Using graphillion %o A339190 from graphillion import GraphSet %o A339190 def make_nXk_king_graph(n, k): %o A339190 grids = [] %o A339190 for i in range(1, k + 1): %o A339190 for j in range(1, n): %o A339190 grids.append((i + (j - 1) * k, i + j * k)) %o A339190 if i < k: %o A339190 grids.append((i + (j - 1) * k, i + j * k + 1)) %o A339190 if i > 1: %o A339190 grids.append((i + (j - 1) * k, i + j * k - 1)) %o A339190 for i in range(1, k * n, k): %o A339190 for j in range(1, k): %o A339190 grids.append((i + j - 1, i + j)) %o A339190 return grids %o A339190 def A339190(n, k): %o A339190 universe = make_nXk_king_graph(n, k) %o A339190 GraphSet.set_universe(universe) %o A339190 cycles = GraphSet.cycles(is_hamilton=True) %o A339190 return cycles.len() %o A339190 print([A339190(j + 2, i - j + 2) for i in range(10 - 1) for j in range(i + 1)]) %Y A339190 Rows and columns 3..5 give A339200, A339201, A339202. %Y A339190 Main diagonal gives A140519. %Y A339190 Cf. A321172, A339098, A339849. %K A339190 nonn,tabl %O A339190 2,1 %A A339190 _Seiichi Manyama_, Nov 27 2020