A340043 Triangle T(n,k), n>=1, 0 <= k <= A002620(n-1), read by rows, where T(n,k) is the number of self-avoiding paths of length 2*(n+k) along the edges of a grid with n X n square cells, which do not pass above the diagonal, start at the lower left corner and finish at the upper right corner.
1, 2, 5, 2, 14, 16, 10, 42, 90, 123, 94, 20, 132, 440, 954, 1460, 1524, 922, 248, 429, 2002, 6017, 13688, 24582, 34536, 35487, 24042, 8852, 1072, 1430, 8736, 33784, 101232, 251646, 530900, 944042, 1369110, 1541774, 1264402, 693740, 221738, 31178
Offset: 1
Examples
Triangle begins:
1;
2;
5, 2;
14, 16, 10;
42, 90, 123, 94, 20;
132, 440, 954, 1460, 1524, 922, 248;
429, 2002, 6017, 13688, 24582, 34536, 35487, 24042, 8852, 1072;
Links
- Siqi Wang, Rows n = 1..24, flattened (rows 1..14 from Seiichi Manyama).
- Siqi Wang, C++ program used to generate the sequence.
Programs
-
Python
# Using graphillion from graphillion import GraphSet def make_stairs(n): s = 1 grids = [] for i in range(n + 1, 1, -1): for j in range(i - 1): a, b, c = s + j, s + j + 1, s + i + j grids.extend([(a, b), (a, c)]) s += i return grids def A340043(n): universe = make_stairs(n) GraphSet.set_universe(universe) start, goal = n + 1, (n + 1) * (n + 2) // 2 paths = GraphSet.paths(start, goal) return [paths.len(2 * (n + k)).len() for k in range((n - 1) * (n - 1) // 4 + 1)] print([i for n in range(1, 9) for i in A340043(n)])