A339849 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of Hamiltonian circuits within parallelograms of size n X k on the triangular lattice.

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 44, 80, 44, 1, 1, 148, 549, 549, 148, 1, 1, 498, 3851, 7104, 3851, 498, 1, 1, 1676, 26499, 104100, 104100, 26499, 1676, 1, 1, 5640, 183521, 1475286, 3292184, 1475286, 183521, 5640, 1, 1, 18980, 1269684, 20842802, 100766213, 100766213, 20842802, 1269684, 18980, 1

Offset

2,5

Links

Seiichi Manyama, Antidiagonals n = 2..13, flattened

M. Peto, Studies of protein designability using reduced models, Thesis, 2007.

Formula

T(n,k) = T(k,n).

Example

Square array T(n,k) begins:
  1,   1,     1,       1,         1,          1, ...
  1,   4,    13,      44,       148,        498, ...
  1,  13,    80,     549,      3851,      26499, ...
  1,  44,   549,    7104,    104100,    1475286, ...
  1, 148,  3851,  104100,   3292184,  100766213, ...
  1, 498, 26499, 1475286, 100766213, 6523266332, ...

Prog

(Python)
# Using graphillion
from graphillion import GraphSet
def make_T_nk(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))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
    return grids
def A339849(n, k):
    universe = make_T_nk(n, k)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles(is_hamilton=True)
    return cycles.len()
print([A339849(j + 2, i - j + 2) for i in range(11 - 1) for j in range(i + 1)])

Crossrefs

Rows and columns 3..10 give A339850, A339851, A339852, A338970, A339622, A339960, A339961, A339962.

Main diagonal gives A339854.

Cf. A339190.

Sequence in context: A212801 A147565 A022167 * A064281 A267318 A050154

Adjacent sequences: A339846 A339847 A339848 * A339850 A339851 A339852

Keyword

nonn,tabl

Author

Seiichi Manyama, Dec 19 2020

Status

approved