A333652
Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-n, read by rows, where T(n,k) is the number of 2*(k+n)-cycles in the n X n grid graph which pass through NW and SW corners.
Original entry on oeis.org
1, 1, 3, 1, 6, 17, 17, 6, 1, 10, 45, 167, 404, 570, 460, 186, 1, 15, 100, 506, 2164, 7726, 20483, 39401, 56015, 57632, 37450, 10340, 1072, 1, 21, 196, 1316, 7066, 33983, 147377, 546400, 1656592, 4099732, 8394433, 14227675, 19443270, 20239262, 14767415, 7007270, 1926990, 230440
Offset: 2
T(3,0) = 1;
+--*
| |
* *
| |
+--*
T(3,1) = 3;
+--*--* +--*--* +--*
| | | | | |
* * * *--* * *--*
| | | | | |
+--*--* +--* +--*--*
Triangle starts:
====================================================================
n\k| 0 1 2 3 4 ... 7 ... 12 ... 17 ... 24
---|----------------------------------------------------------------
2 | 1;
3 | 1, 3;
4 | 1, 6, 17, 17, 6;
5 | 1, 10, 45, 167, 404, ... , 186;
6 | 1, 15, 100, 506, 2164, .......... , 1072;
7 | 1, 21, 196, 1316, 7066, .................. , 230440;
8 | 1, 28, 350, 3038, 20317, ............................ , 4638576;
-
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333652(n):
universe = tl.grid(n - 1, n - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles().including(1).including(n)
return [cycles.len(2 * k).len() for k in range(n, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333652(n)])
A333667
Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2*n+2, read by rows, where T(n,k) is the number of 2*(k+2*n-2)-cycles in the n X n grid graph which pass through NW and SE corners ((0,0),(n-1,n-1)).
Original entry on oeis.org
1, 3, 20, 16, 6, 175, 420, 562, 456, 186, 1764, 8064, 21224, 39500, 55376, 57248, 37586, 10260, 1072, 19404, 138600, 569768, 1717152, 4151965, 8371428, 14126846, 19364732, 20241450, 14759356, 6998166, 1927724, 230440
Offset: 2
T(3,0) = 3;
+--*--* +--*--* +--*
| | | | | |
*--* * * * * *--*
| | | | | |
*--+ *--*--+ *--*--+
Triangle starts:
=======================================================================
n\k| 0 1 2 ... 4 ... 8 ... 12 ... 18
---|-------------------------------------------------------------------
2 | 1;
3 | 3;
4 | 20, 16, 6;
5 | 175, 420, 562, ... , 186;
6 | 1764, 8064, 21224, .......... , 1072;
7 | 19404, 138600, 569768, .................. , 230440;
8 | 226512, 2265120, 12922446, ............................ , 4638576;
-
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333667(n):
universe = tl.grid(n - 1, n - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles().including(1).including(n * n)
return [cycles.len(2 * k).len() for k in range(2 * n - 2, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333667(n)])
A333668
Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2*n+2, read by rows, where T(n,k) is the number of 2*(k+2*n-2)-cycles in the n X n grid graph which pass through four corners ((0,0), (0,n-1), (n-1,n-1), (n-1,0)).
Original entry on oeis.org
1, 1, 1, 4, 6, 1, 12, 58, 156, 146, 1, 24, 244, 1416, 5435, 12976, 16654, 7108, 1072, 1, 40, 696, 7076, 47965, 236628, 873610, 2348664, 4335724, 4958224, 3407276, 1298704, 205792
Offset: 2
T(4,1) = 4;
+--*--*--+ +--*--*--+ +--*--*--+ +--* *--+
| | | | | | | | | |
*--* * * *--* * * * *--* *
| | | | | | | |
*--* * * *--* * *--* * * *
| | | | | | | | | |
+--*--*--+ +--*--*--+ +--* *--+ +--*--*--+
Triangle starts:
=================================================================
n\k| 0 1 2 3 4 ... 8 ... 12 ... 18
---|-------------------------------------------------------------
2 | 1;
3 | 1;
4 | 1, 4, 6;
5 | 1, 12, 58, 156, 146;
6 | 1, 24, 244, 1416, 5435, ... , 1072;
7 | 1, 40, 696, 7076, 47965, ........... , 205792;
8 | 1, 60, 1590, 24960, 263770, ..................... , 4638576;
-
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333668(n):
universe = tl.grid(n - 1, n - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles()
for i in [1, n, n * (n - 1) + 1, n * n]:
cycles = cycles.including(i)
return [cycles.len(2 * k).len() for k in range(2 * n - 2, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333668(n)])
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