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, ...
-
# 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)])
A339199
Number of (undirected) cycles on the n X 5 king graph.
Original entry on oeis.org
204, 33145, 4847163, 545217435, 61575093671, 7050330616441, 808723201743855, 92672075290059017, 10617254793634907021, 1216460857186123433837, 139377550879455782939427, 15969325570952770252910697, 1829698785056144504575785405, 209639263869115933534540710701
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 A339199(n):
return A339098(n, 5)
print([A339199(n) for n in range(2, 20)])
A339852
Number of Hamiltonian circuits within parallelograms of size 5 X n on the triangular lattice.
Original entry on oeis.org
1, 44, 549, 7104, 104100, 1475286, 20842802, 295671198, 4190083085, 59374628434, 841470846944, 11925007688342, 168996943899738, 2394974040514288, 33940795571394262, 480998063196253650, 6816550836218124869, 96601974078400509612, 1369012239935377295854, 19401203058253673198258
Offset: 2
- Seiichi Manyama, Table of n, a(n) for n = 2..500
- M. Peto, Studies of protein designability using reduced models, Thesis, 2007.
- Index entries for linear recurrences with constant coefficients, signature (8,62,384,160,-1628,-11310,9700,-16019,102564, -98380,263340, -429661,174728,-361330,147404,284641,24764,182412,-156248, -138559,14756,14496,-3660,-2640,328,80,-8)
-
# 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 A339852(n):
return A339849(5, n)
print([A339852(n) for n in range(2, 21)])
A358920
Number of (undirected) paths in the 5 X n king graph.
Original entry on oeis.org
10, 7909, 1622015, 329967798, 57533191444, 9454839968415, 1482823362091281, 224616420155224372, 33098477832558055458, 4770920988514661692889, 675419680016870426617489, 94197848411355615226343472
Offset: 1
Showing 1-4 of 4 results.