A268894
Number of Hamiltonian paths in C_n X P_n.
Original entry on oeis.org
1, 4, 144, 4016, 152230, 14557092, 1966154260, 761411682704, 411068703517542, 684434716944151900, 1572754514153890134760, 11579615738168536799184984, 117186519917858266359631481672, 3877921919790491112398750141807648, 176258463464553583688099296874564393850, 26493868301658838913487471166447301509560736
Offset: 1
A338961
Number of (undirected) paths in C_5 X P_n.
Original entry on oeis.org
20, 1285, 39425, 971610, 21272810, 432363395, 8355404595
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)
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 A338961(n):
return B(5, n)
print([A338961(n) for n in range(1, 6)])
A003689
Number of Hamiltonian paths in K_3 X P_n.
Original entry on oeis.org
3, 30, 144, 588, 2160, 7440, 24576, 78912, 248448, 771456, 2371968, 7241856, 21998976, 66586752, 201025920, 605781120, 1823094144, 5481472128, 16470172032, 49464779904, 148508372352, 445764192384, 1337792747904
Offset: 1
- F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
- F. Faase, Counting Hamiltonian cycles in product graphs
- F. Faase, Results from the counting program
- Index entries for linear recurrences with constant coefficients, signature (7,-16,12).
-
[3,30] cat [128*3^(n-2)-(21*n+57)*2^(n-2): n in [3..30]]; // Vincenzo Librandi, Apr 27 2014
-
Join[{3,30},LinearRecurrence[{7,-16,12},{144,588,2160},30]] (* Harvey P. Dale, Apr 26 2014 *)
CoefficientList[Series[3 (1 + 3 x - 6 x^2 + 8 x^3 - 4 x^4)/((1 - 3 x) (1 - 2 x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 27 2014 *)
-
# 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 A003689(n):
return B(3, n)
print([A003689(n) for n in range(1, 21)]) # Seiichi Manyama, Dec 18 2020
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)])
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