A324604 Triangle read by rows: T(n,k) is the number of non-intersecting loops starting at (0,0) on the n X k torus consisting of steps up and to the right, 1 <= k <= n.

2, 2, 4, 2, 7, 22, 2, 13, 66, 258, 2, 25, 189, 898, 3528, 2, 49, 537, 3118, 14545, 87830, 2, 97, 1544, 11462, 75446, 746589, 8295536, 2, 193, 4508, 44990, 447667, 6130779, 76468304, 760941598, 2, 385, 13322, 182638, 2536744, 41919565, 580156255, 7097004356

Offset

1,1

Comments

Conjecture: Second column is A181565.

Links

Table of n, a(n) for n=1..44.

Peter Kagey, Cycles on the torus, Programming Puzzles & Code Golf Stack Exchange.

Example

The T(3,2) = 7 walks on the 3 X 2 torus are:
  (0,0),(0,1),(0,0)
  (0,0),(1,0),(2,0),(0,0)
  (0,0),(0,1),(1,1),(1,0),(2,0),(0,0)
  (0,0),(0,1),(1,1),(2,1),(2,0),(0,0)
  (0,0),(1,0),(1,1),(2,1),(0,1),(0,0)
  (0,0),(1,0),(1,1),(2,1),(2,0),(0,0)
  (0,0),(1,0),(2,0),(2,1),(0,1),(0,0)
Triangle begins:
  2;
  2,   4;
  2,   7,   22;
  2,  13,   66,   258;
  2,  25,  189,   898,   3528;
  2,  49,  537,  3118,  14545,   87830;
  2,  97, 1544, 11462,  75446,  746589,  8295536;
  2, 193, 4508, 44990, 447667, 6130779, 76468304, 760941598;
  ...

Crossrefs

Cf. A181565, A324603.

Sequence in context: A286369 A132801 A321199 * A270366 A072478 A190014

Adjacent sequences: A324601 A324602 A324603 * A324605 A324606 A324607

Keyword

nonn,tabl,walk

Author

Peter Kagey, Mar 09 2019

Extensions

a(34)-a(44) from Tamás Fülöp, Nov 03 2025

Status

approved