A339136
Number of (undirected) cycles in the graph C_3 X P_n.
Original entry on oeis.org
1, 14, 63, 220, 701, 2154, 6523, 19640, 59001, 177094, 531383, 1594260, 4782901, 14348834, 43046643, 129140080, 387420401, 1162261374, 3486784303, 10460353100, 31381059501, 94143178714, 282429536363, 847288609320, 2541865828201, 7625597484854, 22876792454823, 68630377364740
Offset: 1
-
# Using graphillion
from graphillion import GraphSet
def make_CnXPk(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))
grids.append((i + (n - 1) * k, i))
for i in range(1, k * n, k):
for j in range(1, k):
grids.append((i + j - 1, i + j))
return grids
def A339136(n):
universe = make_CnXPk(3, n)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles()
return cycles.len()
print([A339136(n) for n in range(1, 20)])
A339137
Number of (undirected) cycles in the graph C_4 X P_n.
Original entry on oeis.org
1, 28, 225, 1540, 10217, 67388, 444017, 2925140, 19270105, 126946444, 836290209, 5509263332, 36293601737, 239092863324, 1575081964113, 10376232739316, 68355938510649, 450311249502892, 2966534083948417, 19542759549039748, 128742647137776169, 848123272992954492
Offset: 1
-
# Using graphillion
from graphillion import GraphSet
def make_CnXPk(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))
grids.append((i + (n - 1) * k, i))
for i in range(1, k * n, k):
for j in range(1, k):
grids.append((i + j - 1, i + j))
return grids
def A339137(n):
universe = make_CnXPk(4, n)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles()
return cycles.len()
print([A339137(n) for n in range(1, 20)])
A339140
Number of (undirected) cycles in the graph C_n X P_n.
Original entry on oeis.org
6, 63, 1540, 119235, 29059380, 21898886793, 50826232189144, 361947451544923557, 7884768474166076906420, 524518303312357729182869149, 106448798893410608983300257207398, 65866487708413725073741586390176988083, 124207126413825808953168887580780401519104028
Offset: 2
If we represent each vertex with o, used edges with lines and unused edges with dots, and repeat the wraparound edges on left and right, the a(2) = 6 solutions for n = 2 are:
.o-o. -o.o- .o-o. -o.o- -o-o- .o.o.
| | | | | | | | . . . .
.o-o. .o-o. -o.o- -o.o- .o.o. -o-o-
-
# Using graphillion
from graphillion import GraphSet
def make_CnXPk(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))
grids.append((i + (n - 1) * k, i))
for i in range(1, k * n, k):
for j in range(1, k):
grids.append((i + j - 1, i + j))
return grids
def A339140(n):
universe = make_CnXPk(n, n)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles()
return cycles.len()
print([A339140(n) for n in range(3, 7)])
a(2), a(9), a(11) and a(13)-a(18) from
Ed Wynn, Jun 25 2023
A339142
Number of (undirected) cycles in the graph C_5 X P_n.
Original entry on oeis.org
1, 52, 733, 9394, 119235, 1512196, 19177677, 243212478, 3084441599, 39117172360, 496087629441, 6291429718962, 79788500460003, 1011885230273244, 12832823194696645, 162747064808635206, 2063973507784856167, 26175505197898511728, 331960206747350288969, 4209950410912939269210
Offset: 1
-
# Using graphillion
from graphillion import GraphSet
def make_CnXPk(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))
grids.append((i + (n - 1) * k, i))
for i in range(1, k * n, k):
for j in range(1, k):
grids.append((i + j - 1, i + j))
return grids
def A339142(n):
universe = make_CnXPk(5, n)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles()
return cycles.len()
print([A339142(n) for n in range(1, 9)])
More terms from
Ed Wynn, Jun 28 2023
A338297
Number of Hamiltonian paths in C_6 X P_n.
Original entry on oeis.org
6, 228, 4800, 76116, 1094316, 14557092, 183735204, 2230289220, 26275912776, 302338568832, 3412921463352, 37923555328200, 415863933818988, 4509400849281240, 48428461587426108, 515767225814395500, 5452991323044249720, 57282647077608267072, 598324561437126968664, 6217929367753246782612
Offset: 1
-
# Using graphillion
from graphillion import GraphSet
def make_CnXPk(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))
grids.append((i + (n - 1) * k, i))
for i in range(1, k * n, k):
for j in range(1, k):
grids.append((i + j - 1, i + j))
return grids
def A(start, goal, n, k):
universe = make_CnXPk(n, k)
GraphSet.set_universe(universe)
paths = GraphSet.paths(start, goal, is_hamilton=True)
return paths.len()
def B(n, k):
m = k * n
s = 0
for i in range(1, m):
for j in range(i + 1, m + 1):
s += A(i, j, n, k)
return s
def A338297(n):
return B(6, n)
print([A338297(n) for n in range(1, 11)])
Showing 1-5 of 5 results.