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Number of (undirected) Hamiltonian paths in the 3 X n king graph.
6

%I #17 Feb 16 2025 08:34:01

%S 1,48,392,4678,43676,406396,3568906,30554390,254834078,2085479610,

%T 16791859330,133416458104,1048095087616,8154539310958,62918331433308,

%U 481954854686434,3668399080453520,27766093432542984,209120844634276158,1568050593805721822

%N Number of (undirected) Hamiltonian paths in the 3 X n king graph.

%H Andrew Howroyd, <a href="/A339761/b339761.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphPath.html">Graph Path</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (15,-36,-289,708,2617,-1278,-4641,2263,4808,3286,-1422,-3830,-2200, -432,216,216).

%F 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

%o (Python)

%o # Using graphillion

%o from graphillion import GraphSet

%o def make_nXk_king_graph(n, k):

%o grids = []

%o for i in range(1, k + 1):

%o for j in range(1, n):

%o grids.append((i + (j - 1) * k, i + j * k))

%o if i < k:

%o grids.append((i + (j - 1) * k, i + j * k + 1))

%o if i > 1:

%o grids.append((i + (j - 1) * k, i + j * k - 1))

%o for i in range(1, k * n, k):

%o for j in range(1, k):

%o grids.append((i + j - 1, i + j))

%o return grids

%o def A(start, goal, n, k):

%o universe = make_nXk_king_graph(n, k)

%o GraphSet.set_universe(universe)

%o paths = GraphSet.paths(start, goal, is_hamilton=True)

%o return paths.len()

%o def B(n, k):

%o m = k * n

%o s = 0

%o for i in range(1, m):

%o for j in range(i + 1, m + 1):

%o s += A(i, j, n, k)

%o return s

%o def A339761(n):

%o return B(n, 3)

%o print([A339761(n) for n in range(1, 11)])

%Y Row 3 of A350729.

%Y Cf. A003685, A308129, A339751, A339760, A339762, A339763.

%K nonn,changed

%O 1,2

%A _Seiichi Manyama_, Dec 16 2020