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Number of Hamiltonian circuits within parallelograms of size 7 X n on the triangular lattice.
2

%I #29 Dec 26 2020 02:41:53

%S 1,498,26499,1475286,100766213,6523266332,418172485806,26971800950170,

%T 1738936046774850,112060168171247368,7222422644817870197,

%U 465494892350086836970,30001329862709920944426,1933604967243463575726934,124622105764386987040047037,8031972575008760516889720476

%N Number of Hamiltonian circuits within parallelograms of size 7 X n on the triangular lattice.

%H Seiichi Manyama, <a href="/A339622/b339622.txt">Table of n, a(n) for n = 2..150</a>

%H Olga Bodroža-Pantić, Harris Kwong and Milan Pantić, <a href="https://doi.org/10.1016/j.dam.2015.07.028">Some new characterizations of Hamiltonian cycles in triangular grid graphs</a>, Discrete Appl. Math. 201 (2016) 1-13. (a(n) is equal to h6(n-1) defined by this paper)

%H M. Peto, <a href="https://doi.org/10.31274/rtd-180813-17105">Studies of protein designability using reduced models</a>, Thesis, 2007.

%o (Python)

%o # Using graphillion

%o from graphillion import GraphSet

%o def make_T_nk(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 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 A339849(n, k):

%o universe = make_T_nk(n, k)

%o GraphSet.set_universe(universe)

%o cycles = GraphSet.cycles(is_hamilton=True)

%o return cycles.len()

%o def A339622(n):

%o return A339849(7, n)

%o print([A339622(n) for n in range(2, 8)])

%Y Row 7 of A339849.

%Y Cf. A145416.

%K nonn

%O 2,2

%A _Seiichi Manyama_, Dec 25 2020