A007764 -id:A007764 - cp's OEIS frontend

A235389 Number of paths joining opposite corners of an n X n grid with every vertex appearing at most twice in the path.

22, 9482, 112269228, 37515056726494, 355023905438833546724

Offset: 2

Muhammad Kholilurrohman, Jan 09 2014

			The 22 paths in a 2 X 2 grid whose nodes are numbered from 1 to 4 in the natural way are 124, 134, 12124, 12134, 12424, 12434, 13124, 13134, 13424, 13434, 1212434, 1213424, 1213434, 1242134, 1243124, 1243134, 1312424, 1312434, 1313424, 1342124, 1342134, 1343124. - _Giovanni Resta_, Mar 24 2014

Cf. A007764, A239688.

a(5)-a(6) from Muhammad Kholilurrohman, Nov 13 2014

A265132 Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid that move in 3 or fewer cardinal directions.

Original entry on oeis.org

1, 2, 12, 108, 1180, 15300, 234374, 4190872, 86080572, 1999951380, 51874664446, 1486016035944, 46596167540806, 1587429536107688, 58385852010664650, 2305843009058576432, 97322383750732656572, 4371823119475059457716, 208254700595813407930382

Offset: 1

Theodore M. Mishura, Dec 02 2015

This sequence counts all joining paths that move in one of the following ways: UP and RIGHT only; UP, RIGHT, and LEFT only; UP, RIGHT, and DOWN only.

Cf. A007764.

[2*(n+1)^n-Binomial(2*n,n): n in [0..20]]; // Vincenzo Librandi, Dec 03 2015

Table[2 (n + 1)^n - Binomial[2 n, n], {n, 0, 18}] (* Michael De Vlieger, Dec 02 2015 *)

a(n) = 2*(n+1)^n - binomial(2*n,n); \\ Altug Alkan, Dec 03 2015

a(n) = 2*(n+1)^n - C(2*n,n).

A363577 Number of inequivalent Hamiltonian paths starting in the lower left corner of an n X n grid graph (paths differing only by rotations and reflections are regarded as equivalent).

Original entry on oeis.org

1, 1, 3, 23, 347, 10199, 683227, 85612967, 25777385143, 14396323278040, 19799561204761862, 50351228336401026361, 319210377672595552740369, 3736517399241599771428011100, 109790442395888863208285555153329, 5952238893391106787883489313797219949

Offset: 1

Lars Blomberg, Jun 10 2023

Equivalently, number of inequivalent Hamiltonian paths starting in a corner of an n X n grid graph (paths differing only by rotations and reflections are regarded as equivalent). - Martin Ehrenstein, Jul 08 2023

			There are 3 paths for n=3:
  +--+--+    +--+--+    +--+  +
  |     |    |     |    |  |  |
  +  +  +    +  +--+    +  +  +
  |  |  |    |  |       |  |  |
  +  +--+    +  +--+    +  +--+
A fourth path:
  +--+--+
        |
  +--+  +
  |  |  |
  +  +--+
is the same as the second one in the row above after a 90-degree rotation.
All paths starting E are the same as the corresponding ones starting N after reflection in the forward diagonal.

Oliver R. Bellwood, Heitor P. Casagrande, and William J. Munro, Fractal Path Strategies for Efficient 2D DMRG Simulations, arXiv:2507.11820 [cond-mat.str-el], 2025. See p. 4.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Hamiltonian Path
Index entries for sequences related to graphs, Hamiltonian

Cf. A000532, A001184, A003763, A007764, A120443, A121785, A121789, A145157, A271507, A384173.

a(1) added by N. J. A. Sloane, Jun 10 2023
a(8)-a(9) from Martin Ehrenstein, Jul 08 2023
a(10)-a(16) from Oliver R. Bellwood, Jun 06 2025

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A235389 Number of paths joining opposite corners of an n X n grid with every vertex appearing at most twice in the path.

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A265132 Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid that move in 3 or fewer cardinal directions.

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A363577 Number of inequivalent Hamiltonian paths starting in the lower left corner of an n X n grid graph (paths differing only by rotations and reflections are regarded as equivalent).

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