A339190
Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of (undirected) Hamiltonian cycles on the n X k king graph.
Original entry on oeis.org
3, 4, 4, 8, 16, 8, 16, 120, 120, 16, 32, 744, 2830, 744, 32, 64, 4922, 50354, 50354, 4922, 64, 128, 31904, 1003218, 2462064, 1003218, 31904, 128, 256, 208118, 19380610, 139472532, 139472532, 19380610, 208118, 256, 512, 1354872, 378005474, 7621612496, 22853860116, 7621612496, 378005474, 1354872, 512
Offset: 2
Square array T(n,k) begins:
3, 4, 8, 16, 32, 64, ...
4, 16, 120, 744, 4922, 31904, ...
8, 120, 2830, 50354, 1003218, 19380610, ...
16, 744, 50354, 2462064, 139472532, 7621612496, ...
32, 4922, 1003218, 139472532, 22853860116, 3601249330324, ...
64, 31904, 19380610, 7621612496, 3601249330324, 1622043117414624, ...
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# Using graphillion
from graphillion import GraphSet
def make_nXk_king_graph(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))
if i > 1:
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 A339190(n, k):
universe = make_nXk_king_graph(n, k)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles(is_hamilton=True)
return cycles.len()
print([A339190(j + 2, i - j + 2) for i in range(10 - 1) for j in range(i + 1)])
A339197
Number of (undirected) cycles on the n X 3 king graph.
Original entry on oeis.org
30, 348, 3459, 33145, 316164, 3013590, 28722567, 273751765, 2609096478, 24866992602, 237004387635, 2258860992595, 21528938911842, 205189789087374, 1955639788756293, 18638973217791295, 177645865363829526, 1693121885638023396, 16136945905019298321, 153799336805212613275
Offset: 2
-
# Using graphillion
from graphillion import GraphSet
def make_nXk_king_graph(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))
if i > 1:
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 A339098(n, k):
universe = make_nXk_king_graph(n, k)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles()
return cycles.len()
def A339197(n):
return A339098(n, 3)
print([A339197(n) for n in range(2, 30)])
A339850
Number of Hamiltonian circuits within parallelograms of size 3 X n on the triangular lattice.
Original entry on oeis.org
1, 4, 13, 44, 148, 498, 1676, 5640, 18980, 63872, 214944, 723336, 2434192, 8191616, 27566672, 92768192, 312186304, 1050578720, 3535439040, 11897565568, 40038044736, 134737229824, 453421769728, 1525868548224, 5134898635008, 17280115002368, 58151561641216
Offset: 2
a(2) = 1:
*---*
/ /
* *
/ /
*---*
a(3) = 4:
* *---* *---*---*
/ \ / / \ /
* * * *---* *
/ / / /
*---*---* *---*---*
*---*---* *---*---*
/ / / /
* * * * *---*
/ / \ / / \
*---* * *---*---*
- Seiichi Manyama, Table of n, a(n) for n = 2..1000
- Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
- M. Peto, Studies of protein designability using reduced models, Thesis, 2007.
- Index entries for linear recurrences with constant coefficients, signature (2,4,2).
-
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 *)
-
my(N=66, x='x+O('x^N)); Vec((x*(1+x))^2/(1-2*x-4*x^2-2*x^3))
-
# 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 A339850(n):
return A339849(3, n)
print([A339850(n) for n in range(2, 21)])
Showing 1-3 of 3 results.