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 A339761 Number of (undirected) Hamiltonian paths in the 3 X n king graph. 6
 1, 48, 392, 4678, 43676, 406396, 3568906, 30554390, 254834078, 2085479610, 16791859330, 133416458104, 1048095087616, 8154539310958, 62918331433308, 481954854686434, 3668399080453520, 27766093432542984, 209120844634276158, 1568050593805721822 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 Eric Weisstein's World of Mathematics, Graph Path Eric Weisstein's World of Mathematics, King Graph Index entries for linear recurrences with constant coefficients, signature (15,-36,-289,708,2617,-1278,-4641,2263,4808,3286,-1422,-3830,-2200, -432,216,216). FORMULA G.f.: x*(1 + 33*x - 292*x^2 + 815*x^3 + 782*x^4 - 3649*x^5 - 4630*x^6 + 1517*x^7 + 3835*x^8 - 3822*x^9 - 5722*x^10 - 5418*x^11 - 7562*x^12 - 4808*x^13 - 240*x^14 + 720*x^15 + 216*x^16)/((1 - x)*(1 - 4*x - 15*x^2 - 8*x^3 - 6*x^4)^2*(1 - 6*x - 12*x^2 + 27*x^3 - 2*x^4 - 30*x^5 - 4*x^6 + 6*x^7)). - Andrew Howroyd, Jan 17 2022 PROG (Python) # Using graphillion from graphillion import GraphSet def make_nXk_king_graph(n, k): grids = [] for i in range(1, k + 1): for j in range(1, n): grids.append((i + (j - 1) * k, i + j * k)) if i < k: grids.append((i + (j - 1) * k, i + j * k + 1)) if i > 1: grids.append((i + (j - 1) * k, i + j * k - 1)) for i in range(1, k * n, k): for j in range(1, k): grids.append((i + j - 1, i + j)) return grids def A(start, goal, n, k): universe = make_nXk_king_graph(n, k) GraphSet.set_universe(universe) paths = GraphSet.paths(start, goal, is_hamilton=True) return paths.len() def B(n, k): m = k * n s = 0 for i in range(1, m): for j in range(i + 1, m + 1): s += A(i, j, n, k) return s def A339761(n): return B(n, 3) print([A339761(n) for n in range(1, 11)]) CROSSREFS Row 3 of A350729. Cf. A003685, A308129, A339751, A339760, A339762, A339763. Sequence in context: A348120 A160335 A019558 * A232917 A172201 A249293 Adjacent sequences: A339758 A339759 A339760 * A339762 A339763 A339764 KEYWORD nonn AUTHOR Seiichi Manyama, Dec 16 2020 STATUS approved

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Last modified June 20 21:09 EDT 2024. Contains 373532 sequences. (Running on oeis4.)