%I #32 Jul 07 2021 02:01:41
%S 1,4,13,44,148,498,1676,5640,18980,63872,214944,723336,2434192,
%T 8191616,27566672,92768192,312186304,1050578720,3535439040,
%U 11897565568,40038044736,134737229824,453421769728,1525868548224,5134898635008,17280115002368,58151561641216
%N Number of Hamiltonian circuits within parallelograms of size 3 X n on the triangular lattice.
%H Seiichi Manyama, <a href="/A339850/b339850.txt">Table of n, a(n) for n = 2..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H M. Peto, <a href="https://doi.org/10.31274/rtd-180813-17105">Studies of protein designability using reduced models</a>, Thesis, 2007.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,2).
%F G.f.: (x*(1+x))^2/(1-2*x-4*x^2-2*x^3).
%F a(n) = 2*a(n-1) + 4*a(n-2) + 2*a(n-3) for n > 4.
%e a(2) = 1:
%e *---*
%e / /
%e * *
%e / /
%e *---*
%e a(3) = 4:
%e * *---* *---*---*
%e / \ / / \ /
%e * * * *---* *
%e / / / /
%e *---*---* *---*---*
%e *---*---* *---*---*
%e / / / /
%e * * * * *---*
%e / / \ / / \
%e *---* * *---*---*
%t Drop[CoefficientList[Series[(x (1 + x))^2/(1 - 2 x - 4 x^2 - 2 x^3), {x, 0, 28}], x], 2] (* _Michael De Vlieger_, Jul 06 2021 *)
%o (PARI) my(N=66, x='x+O('x^N)); Vec((x*(1+x))^2/(1-2*x-4*x^2-2*x^3))
%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 A339850(n):
%o return A339849(3, n)
%o print([A339850(n) for n in range(2, 21)])
%Y Row 3 of A339849.
%Y Cf. A339200.
%K nonn
%O 2,2
%A _Seiichi Manyama_, Dec 19 2020