A145416 -id:A145416 - cp's OEIS frontend

A321172 Triangle read by rows: T(m,n) = number of Hamiltonian cycles on m X n grid of points (m >= 2, 2 <= n <= m).

1, 1, 0, 1, 2, 6, 1, 0, 14, 0, 1, 4, 37, 154, 1072, 1, 0, 92, 0, 5320, 0, 1, 8, 236, 1696, 32675, 301384, 4638576, 1, 0, 596, 0, 175294, 0, 49483138, 0, 1, 16, 1517, 18684, 1024028, 17066492, 681728204, 13916993782, 467260456608

Offset: 2

Robert FERREOL, Jan 10 2019

Orientation of the path is not important; you can start going either clockwise or counterclockwise. Paths related by symmetries are considered distinct.
The m X n grid of points when drawn forms a (m-1) X (n-1) rectangle of cells, so m-1 and n-1 are sometimes used as indices instead of m and n (see, e. g., Kreweras' reference below).
These cycles are also called "closed non-intersecting rook's tours" on m X n chess board.

			T(5,4)=14 is illustrated in the links above.
Table starts:
=================================================================
m\n|  2    3      4       5         6           7            8
---|-------------------------------------------------------------
2  |  1    1      1       1         1           1            1
3  |  1    0      2       0         4           0            8
4  |  1    2      6      14        37          92          236
5  |  1    0     14       0       154           0         1696
6  |  1    4     37     154      1072        5320        32675
7  |  1    0     92       0      5320           0       301384
8  |  1    8    236    1696     32675      301384      4638576
The table is symmetric, so it is completely described by its lower-left half.

Huaide Cheng, Rows n=2..17 of triangle, flattened
Robert Ferréol, The T(4,5)=14 hamiltonian cycles on 4 X 5 square grid of points; the T(5,6) = 154 cycles on 5 X 6 grid are reduced to 44 different forms.
Germain Kreweras, Dénombrement des cycles hamiltoniens dans un rectangle quadrillé, European Journal of Combinatorics, Volume 13, Issue 6 (1992), page 476.
Robert Stoyan and Volker Strehl, Enumeration of Hamiltonian Circuits in Rectangular Grids, Séminaire Lotharingien de Combinatoire, B34f (1995), 21 pp.
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.

Row/column k=4..12 are: (with interspersed zeros for odd k): A006864, A006865, A145401, A145416, A145418, A160149, A180504, A180505, A213813.
Cf. A003763 (bisection of main diagonal), A222200 (subdiagonal), A231829, A270273, A332307.
T(n,2n) gives A333864.

# Program due to Laurent Jouhet-Reverdy
def cycle(m, n):
     if (m%2==1 and n%2==1): return 0
     grid = [[0]*n for _ in range(m)]
     grid[0][0] = 1; grid[1][0] = 1
     counter = [0]; stop = m*n-1
     def run(i, j, nb_points):
         for ni, nj in [(i-1, j), (i+1, j), (i, j+1), (i, j-1)] :
             if  0<=ni<=m-1 and 0<=nj<=n-1 and grid[ni][nj]==0 and (ni,nj)!=(0,1):
                 grid[ni][nj] = 1
                 run(ni, nj, nb_points+1)
                 grid[ni][nj] = 0
             elif (ni,nj)==(0,1) and nb_points==stop:
                 counter[0] += 1
     run(1, 0, 2)
     return counter[0]
L=[];n=7#maximum for a time < 1 mn
for i in range(2,n):
    for j in range(2,i+1):
       L.append(cycle(i,j))
print(L)

T(m,n) = T(n,m).
T(2m+1,2n+1) = 0.
T(2n,2n) = A003763(n).
T(n,n+1) = T(n+1,n) = A222200(n).
G. functions , G_m(x)=Sum {n>=0} T(m,n-2)*x^n after Stoyan's link p. 18 :
G_2(x) = 1/(1-x) = 1+x+x^2+...
G_3(x) = 1/(1-2*x^2) = 1+2*x^2+4*x^4+...
G_4(x) = 1/(1-2*x-2*x^2+2*x^3-x^4) = 1+2*x+6*x^2+...
G_5(x) = (1+3*x^2)/(1-11*x^2-2*x^6) = 1+14*x^2+154*x^4+...

More terms from Pontus von Brömssen, Feb 15 2021

A333864 Number of Hamiltonian cycles on an n X 2*n grid.

Original entry on oeis.org

1, 4, 236, 18684, 32463802, 54756073582, 2365714170297014, 87106950271042689032, 88514516642574170326003422, 71598455565101470929617326988084, 1673219200189416324422979402201514800461, 29815394539834813572600735261571894552950941626, 15836807024750749574106724392556189684881848226515147589

Offset: 2

Seiichi Manyama, Apr 08 2020

nonn

Huaide Cheng, Table of n, a(n) for n = 2..16
Olga Bodroža-Pantić, B. Pantić, I. Pantić AND M. Bodroža-Solarov: Enumeration of Hamiltonian cycles in some grid grafs. MATCH Commun. Math. Comput. Chem. 70:1 (2013), 181-204. on Research Gate.

Cf. A003763, A005390, A145416, A160149, A180505, A321172, A333863.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333864(n):
    universe = tl.grid(n - 1, 2 * n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles(is_hamilton=True)
    return cycles.len()
print([A333864(n) for n in range(2, 8)])

a(n) = A321172(n,2*n).

a(10) and a(12) quoted from Olga's paper.

a(14) from Huaide Cheng, Jul 02 2025

A339622 Number of Hamiltonian circuits within parallelograms of size 7 X n on the triangular lattice.

Original entry on oeis.org

1, 498, 26499, 1475286, 100766213, 6523266332, 418172485806, 26971800950170, 1738936046774850, 112060168171247368, 7222422644817870197, 465494892350086836970, 30001329862709920944426, 1933604967243463575726934, 124622105764386987040047037, 8031972575008760516889720476

Offset: 2

Seiichi Manyama, Dec 25 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 2..150
Olga Bodroža-Pantić, Harris Kwong and Milan Pantić, Some new characterizations of Hamiltonian cycles in triangular grid graphs, Discrete Appl. Math. 201 (2016) 1-13. (a(n) is equal to h6(n-1) defined by this paper)
M. Peto, Studies of protein designability using reduced models, Thesis, 2007.

Row 7 of A339849.

Cf. A145416.

# 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()
def A339622(n):
    return A339849(7, n)
print([A339622(n) for n in range(2, 8)])

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A321172 Triangle read by rows: T(m,n) = number of Hamiltonian cycles on m X n grid of points (m >= 2, 2 <= n <= m).

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A333864 Number of Hamiltonian cycles on an n X 2*n grid.

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A339622 Number of Hamiltonian circuits within parallelograms of size 7 X n on the triangular lattice.

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A222195 Order of linear recurrence for number of Hamiltonian cycles in the graph P_n X P_{2k} (n odd) or P_n X P_k (n even), as a function of k.

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