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A339750 Number of (undirected) paths in the 2 X n king graph.

%I A339750 #21 Feb 16 2025 08:34:01
%S A339750 1,30,235,1448,7909,40674,202719,994268,4837337,23441366,113377235,
%T A339750 547864528,2646278093,12779454410,61709221831,297968336836,
%U A339750 1438739595201,6946894643134,33542671171515,161958548471736,782005482553269,3775857399168946,18231454211243951,88029252078796716
%N A339750 Number of (undirected) paths in the 2 X n king graph.
%H A339750 Seiichi Manyama, <a href="/A339750/b339750.txt">Table of n, a(n) for n = 1..50</a>
%H A339750 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphPath.html">Graph Path</a>
%H A339750 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>
%F A339750 Empirical g.f.: x*(16*x^4 - 48*x^3 + 32*x^2 - 20*x - 1) / ((x-1)^2 * (2*x - 1)^2 * (4*x^2 + 4*x - 1)). - _Vaclav Kotesovec_, Dec 16 2020
%o A339750 (Python)
%o A339750 # Using graphillion
%o A339750 from graphillion import GraphSet
%o A339750 def make_nXk_king_graph(n, k):
%o A339750     grids = []
%o A339750     for i in range(1, k + 1):
%o A339750         for j in range(1, n):
%o A339750             grids.append((i + (j - 1) * k, i + j * k))
%o A339750             if i < k:
%o A339750                 grids.append((i + (j - 1) * k, i + j * k + 1))
%o A339750             if i > 1:
%o A339750                 grids.append((i + (j - 1) * k, i + j * k - 1))
%o A339750     for i in range(1, k * n, k):
%o A339750         for j in range(1, k):
%o A339750             grids.append((i + j - 1, i + j))
%o A339750     return grids
%o A339750 def A(start, goal, n, k):
%o A339750     universe = make_nXk_king_graph(n, k)
%o A339750     GraphSet.set_universe(universe)
%o A339750     paths = GraphSet.paths(start, goal)
%o A339750     return paths.len()
%o A339750 def A307026(n, k):
%o A339750     m = k * n
%o A339750     s = 0
%o A339750     for i in range(1, m):
%o A339750         for j in range(i + 1, m + 1):
%o A339750             s += A(i, j, n, k)
%o A339750     return s
%o A339750 def A339750(n):
%o A339750     return A307026(n, 2)
%o A339750 print([A339750(n) for n in range(1, 21)])
%Y A339750 Row 2 of A307026.
%Y A339750 Cf. A288516, A339760.
%K A339750 nonn
%O A339750 1,2
%A A339750 _Seiichi Manyama_, Dec 15 2020