A324605 Triangle read by rows: T(n,k) is the number of maximal-length 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, 1, 2, 1, 5, 6, 1, 2, 11, 8, 1, 9, 14, 19, 30, 1, 2, 3, 2, 29, 12, 1, 13, 76, 27, 99, 41, 126, 1, 2, 23, 8, 171, 2, 55, 128, 1, 17, 6, 35, 44, 3, 62, 71

Offset

1,1

Comments

Conjecture: T(2*n,2) = 2 for all n.

Proof: The loop must go right after going up in order to not miss any points, forcing a staircase-like pattern that is fully determined by the first step. - Tamás Fülöp, Dec 03 2025

Conjecture: T(2*n+1,2) = 4*n+1 for all n.

Conjecture: T(n,n) = A056188(n) for n > 1.

Conjecture: T(n,n-1) = A306779(n,n-1) = n*(n-1)-1 for n > 1. - Tamás Fülöp, Nov 15 2025

Conjecture: T(2*n,2*k) is even for all n, k. - Tamás Fülöp, Nov 21 2025

Links

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

Formula

T(n,1) = 1 for n > 1.

T(n,n) is even for all n.

T(2*n,2) = 2 for all n. - Tamás Fülöp, Dec 03 2025

Example

Table begins:
  2;
  1,  2;
  1,  5,  6;
  1,  2, 11,  8;
  1,  9, 14, 19,  30;
  1,  2,  3,  2,  29, 12;
  1, 13, 76, 27,  99, 41, 126;
  1,  2, 23,  8, 171,  2,  55, 128;

Crossrefs

Cf. A306779.

Sequence in context: A102551 A217437 A152823 * A086545 A337395 A126083

Adjacent sequences: A324602 A324603 A324604 * A324606 A324607 A324608

Keyword

nonn,tabl,more,walk

Author

Peter Kagey, Mar 09 2019

Extensions

a(34)-a(44) from Tamás Fülöp, Nov 15 2025

Status

approved