A324604 -id:A324604 - cp's OEIS frontend

A306779 Table read by rows: T(n,k) is the length of the longest non-intersecting loop starting at (0,0) on the n X k torus consisting of steps up and to the right, 1 <= k <= n.

1, 2, 4, 3, 5, 9, 4, 8, 11, 16, 5, 9, 14, 19, 25, 6, 12, 18, 24, 29, 36, 7, 13, 19, 27, 33, 41

Offset: 1

Peter Kagey, Mar 09 2019

Conjecture: T(n,k) = n*k if and only if k = 1 or gcd(n,k) > 1.

			Table begins:
  1;
  2,  4;
  3,  5,  9;
  4,  8, 11, 16;
  5,  9, 14, 19, 25;
  6, 12, 18, 24, 29, 36;

Cf. A324604.

A324603 Number of non-intersecting loops starting at (0,0) on the n X n torus consisting of steps up and to the right.

Original entry on oeis.org

2, 4, 22, 258, 3528, 87830, 8295536

Offset: 1

Peter Kagey, Mar 09 2019

			On the 2 X 2 torus, the 4 walks are:
  [(0,0),(1,0),(0,0)]
  [(0,0),(0,1),(0,0)]
  [(0,0),(0,1),(1,1),(1,0),(0,0)]
  [(0,0),(1,0),(1,1),(0,1),(0,0)]
On the 3 X 3 torus, the 22 walks are all equivalent to the following eleven walks by switching coordinates:
  [(0,0),(0,1),(0,2),(0,0)]
  [(0,0),(0,1),(1,1),(2,1),(2,2),(0,2),(0,0)]
  [(0,0),(0,1),(1,1),(2,1),(2,2),(2,0),(0,0)]
  [(0,0),(0,1),(1,1),(1,2),(2,2),(0,2),(0,0)]
  [(0,0),(0,1),(1,1),(1,2),(2,2),(2,0),(0,0)]
  [(0,0),(0,1),(1,1),(1,2),(1,0),(2,0),(0,0)]
  [(0,0),(0,1),(0,2),(1,2),(2,2),(2,0),(0,0)]
  [(0,0),(0,1),(0,2),(1,2),(1,0),(2,0),(0,0)]
  [(0,0),(0,1),(1,1),(2,1),(2,2),(0,2),(1,2),(1,0),(2,0),(0,0)]
  [(0,0),(0,1),(1,1),(1,2),(1,0),(2,0),(2,1),(2,2),(0,2),(0,0)]
  [(0,0),(0,1),(0,2),(1,2),(1,0),(1,1),(2,1),(2,2),(2,0),(0,0)]

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

Cf. A000984, A324604.

cp's OEIS Frontend

A306779 Table read by rows: T(n,k) is the length of the longest non-intersecting loop starting at (0,0) on the n X k torus consisting of steps up and to the right, 1 <= k <= n.

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