A333466 -id:A333466 - cp's OEIS frontend

A333513 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of self-avoiding closed paths on an n X k grid which pass through four corners ((0,0), (0,k-1), (n-1,k-1), (n-1,0)).

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 7, 11, 7, 1, 1, 17, 49, 49, 17, 1, 1, 41, 229, 373, 229, 41, 1, 1, 99, 1081, 3105, 3105, 1081, 99, 1, 1, 239, 5123, 26515, 44930, 26515, 5123, 239, 1, 1, 577, 24323, 227441, 674292, 674292, 227441, 24323, 577, 1

Offset: 2

Seiichi Manyama, Mar 25 2020

			Square array T(n,k) begins:
  1,  1,    1,     1,      1,        1, ...
  1,  1,    3,     7,     17,       41, ...
  1,  3,   11,    49,    229,     1081, ...
  1,  7,   49,   373,   3105,    26515, ...
  1, 17,  229,  3105,  44930,   674292, ...
  1, 41, 1081, 26515, 674292, 17720400, ...

Seiichi Manyama, Antidiagonals n = 2..15, flattened

Column k=2-7 give: A000012, A001333(n-2), A333514, A333515, A358712, A358713.
Main diagonal gives A333466.
Cf. A333758.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333513(n, k):
    universe = tl.grid(n - 1, k - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    for i in [1, k, k * (n - 1) + 1, k * n]:
        cycles = cycles.including(i)
    return cycles.len()
print([A333513(j + 2, i - j + 2) for i in range(11 - 1) for j in range(i + 1)])

T(n,k) = T(k,n).

A333246 Number of self-avoiding closed paths on an n X n grid which pass through NW corner.

Original entry on oeis.org

1, 7, 97, 4111, 532269, 212372937, 263708907211, 1013068026356375, 11955420069208095719, 432101605951906251627393, 47778407166747833830058004149, 16149888968763663448192636077980753, 16675786862526496319891707194153887550751, 52568166380872328447478940416604864445574575709

Offset: 2

Seiichi Manyama, Mar 23 2020

nonn

			a(2) = 1;
   +--*
   |  |
   *--*
a(3) = 7;
   +--*      +--*--*   +--*--*   +--*
   |  |      |     |   |     |   |  |
   *--*      *--*--*   *     *   *  *
                       |     |   |  |
                       *--*--*   *--*
   +--*--*   +--*--*   +--*
   |     |   |     |   |  |
   *  *--*   *--*  *   *  *--*
   |  |         |  |   |     |
   *--*         *--*   *--*--*

Cf. A140517, A333247, A333323, A333438, A333439, A333466.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333246(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles().including(1)
    return cycles.len()
print([A333246(n) for n in range(2, 10)])

a(n) = A333439(n) - 1 for n > 1.

a(11), a(13) from Seiichi Manyama, Apr 07 2020

a(10), a(12), a(14)-a(15) from Andrew Howroyd, Jan 30 2022

A333323 Number of self-avoiding closed paths on an n X n grid which pass through NW and SE corners.

Original entry on oeis.org

1, 3, 42, 1799, 232094, 92617031, 115156685746, 442641690778179, 5224287477491915786, 188825256606226776728029, 20879416139356164466643759334, 7057757437924198729598570424130207, 7287699030020917172151307665469211016474, 22973720258279267139936821063450448822110219653

Offset: 2

Seiichi Manyama, Mar 23 2020

nonn

			a(2) = 1;
   +--*
   |  |
   *--+
a(3) = 3;
   +--*--*   +--*--*   +--*
   |     |   |     |   |  |
   *--*  *   *     *   *  *--*
      |  |   |     |   |     |
      *--+   *--*--+   *--*--+

Anthony J. Guttmann and Iwan Jensen, Table of n, a(n) for n = 2..27
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D2 (with offset 1).
Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.

Cf. A007764, A333246, A333247, A333466.

Cf. A121785, A356610-A356616, A354511.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333323(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles().including(1).including(n * n)
    return cycles.len()
print([A333323(n) for n in range(2, 10)])

a(11) from Seiichi Manyama, Apr 07 2020

a(10) and a(12)-a(15) from Vaclav Kotesovec, Aug 16 2022 (computed by Anthony Guttmann)

A333247 Number of self-avoiding closed paths on an n X n grid which pass through NW and SW corners.

Original entry on oeis.org

1, 4, 47, 1843, 232905, 92729439, 115234959344, 442748883422394

Offset: 2

Seiichi Manyama, Mar 23 2020

a(11) = 188829168009674568016545. - Seiichi Manyama, Apr 07 2020

			a(2) = 1;
   +--*
   |  |
   +--*
a(3) = 4;
   +--*--*   +--*--*   +--*      +--*
   |     |   |     |   |  |      |  |
   *     *   *  *--*   *  *--*   *  *
   |     |   |  |      |     |   |  |
   +--*--*   +--*      +--*--*   +--*

Cf. A271507, A333246, A333323, A333466.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333247(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles().including(1).including(n)
    return cycles.len()
print([A333247(n) for n in range(2, 10)])

A333668 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2n+2, read by rows, where T(n,k) is the number of 2(k+2*n-2)-cycles in the n X n grid graph which pass through four corners ((0,0), (0,n-1), (n-1,n-1), (n-1,0)).

Original entry on oeis.org

1, 1, 1, 4, 6, 1, 12, 58, 156, 146, 1, 24, 244, 1416, 5435, 12976, 16654, 7108, 1072, 1, 40, 696, 7076, 47965, 236628, 873610, 2348664, 4335724, 4958224, 3407276, 1298704, 205792

Offset: 2

Seiichi Manyama, Apr 01 2020

			T(4,1) = 4;
   +--*--*--+   +--*--*--+   +--*--*--+   +--*  *--+
   |        |   |        |   |        |   |  |  |  |
   *--*     *   *     *--*   *        *   *  *--*  *
      |     |   |     |      |        |   |        |
   *--*     *   *     *--*   *  *--*  *   *        *
   |        |   |        |   |  |  |  |   |        |
   +--*--*--+   +--*--*--+   +--*  *--+   +--*--*--+
Triangle starts:
=================================================================
n\k| 0   1     2      3       4 ...       8 ...    12 ...     18
---|-------------------------------------------------------------
2  | 1;
3  | 1;
4  | 1,  4,    6;
5  | 1, 12,   58,   156,    146;
6  | 1, 24,  244,  1416,   5435, ... , 1072;
7  | 1, 40,  696,  7076,  47965, ........... , 205792;
8  | 1, 60, 1590, 24960, 263770, ..................... , 4638576;

Seiichi Manyama, Rows n = 2..9, flattened

Row sums give A333466.

Cf. A003763, A046092, A302337, A333651, A333652, A333667.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333668(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    for i in [1, n, n * (n - 1) + 1, n * n]:
        cycles = cycles.including(i)
    return [cycles.len(2 * k).len() for k in range(2 * n - 2, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333668(n)])

T(n,0) = 1.

T(n,1) = A046092(n-3).

A333759 Number of self-avoiding closed paths in the n X n grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.

Original entry on oeis.org

1, 1, 11, 191, 11346, 2002405, 1112939654, 1878223479450

Offset: 2

Seiichi Manyama, Apr 04 2020

a(11) = 152567999801505122456.

			a(2) = 1;
   +--+
   |  |
   +--+
a(3) = 1;
   +--+--+
   |     |
   +     +
   |     |
   +--+--+
a(4) = 11;
   +--+--+--+   +--+--+--+   +--+--+--+
   |        |   |        |   |        |
   +--*--*  +   +--*  *--+   +--*     +
         |  |      |  |         |     |
   +--*--*  +   +--*  *--+   +--*     +
   |        |   |        |   |        |
   +--+--+--+   +--+--+--+   +--+--+--+
   +--+--+--+   +--+--+--+   +--+--+--+
   |        |   |        |   |        |
   +  *--*--+   +  *--*  +   +     *--+
   |  |         |  |  |  |   |     |
   +  *--*--+   +  *  *  +   +     *--+
   |        |   |  |  |  |   |        |
   +--+--+--+   +--+  +--+   +--+--+--+
   +--+--+--+   +--+--+--+   +--+  +--+
   |        |   |        |   |  |  |  |
   +        +   +        +   +  *--*  +
   |        |   |        |   |        |
   +  *--*  +   +        +   +  *--*  +
   |  |  |  |   |        |   |  |  |  |
   +--+  +--+   +--+--+--+   +--+  +--+
   +--+  +--+   +--+  +--+
   |  |  |  |   |  |  |  |
   +  *--*  +   +  *  *  +
   |        |   |  |  |  |
   +        +   +  *--*  +
   |        |   |        |
   +--+--+--+   +--+--+--+

Main diagonal of A333758.

Cf. A333466.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333759(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    points = [i for i in range(1, n * n + 1) if i % n < 2 or ((i - 1) // n + 1) % n < 2]
    for i in points:
        cycles = cycles.including(i)
    return cycles.len()
print([A333759(n) for n in range(2, 10)])

A333795 Number of self-avoiding closed paths on an n X n grid which pass through all points on the two diagonals of the grid.

Original entry on oeis.org

1, 0, 6, 68, 6102, 1404416, 1094802826, 2524252113468

Offset: 2

Seiichi Manyama, Apr 05 2020

a(11) = 407977071391342237828.

			a(2) = 1;
   +--+
   |  |
   +--+
a(4) = 6;
   +--*--*--+   +--*--*--+   +--*--*--+
   |        |   |        |   |        |
   *--+--+  *   *--+  +--*   *  +--+--*
         |  |      |  |      |  |
   *--+--+  *   *--+  +--*   *  +--+--*
   |        |   |        |   |        |
   +--*--*--+   +--*--*--+   +--*--*--+
   +--*--*--+   +--*  *--+   +--*  *--+
   |        |   |  |  |  |   |  |  |  |
   *  +--+  *   *  +--+  *   *  +  +  *
   |  |  |  |   |        |   |  |  |  |
   *  +  +  *   *  +--+  *   *  +--+  *
   |  |  |  |   |  |  |  |   |        |
   +--*  *--+   +--*  *--+   +--*--*--+

Cf. A333455, A333464, A333466, A333796.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333795(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    points = [i + 1 for i in range(n * n) if i % n - i // n == 0 or i % n + i // n == n - 1]
    for i in points:
        cycles = cycles.including(i)
    return cycles.len()
print([A333795(n) for n in range(2, 10)])

A333796 Number of self-avoiding closed paths on an n X n grid which pass through all points on the diagonal connecting NW and SE corners.

Original entry on oeis.org

1, 2, 22, 716, 73346, 23374544, 23037365786, 69630317879888

Offset: 2

Seiichi Manyama, Apr 05 2020

a(11) = 18267559028025887599256.

			a(2) = 1;
   +--*
   |  |
   *--+
a(3) = 2;
   +--*--*   +--*
   |     |   |  |
   *--+  *   *  +--*
      |  |   |     |
      *--+   *--*--+
a(4) = 22;
   +--*--*--*   +--*--*--*   +--*--*--*
   |        |   |        |   |        |
   *--+--*  *   *--+--*  *   *--+--*  *
         |  |         |  |         |  |
   *--*--+  *      *--+  *         +  *
   |        |      |     |         |  |
   *--*--*--+      *--*--+         *--+
   +--*--*--*   +--*--*--*   +--*--*--*
   |        |   |        |   |        |
   *--+  *--*   *--+  *--*   *--+     *
      |  |         |  |         |     |
   *--*  +--*      *  +--*      *--+  *
   |        |      |     |         |  |
   *--*--*--+      *--*--+         *--+
   +--*--*--*   +--*--*--*   +--*--*--*
   |        |   |        |   |        |
   *  +--*--*   *  +--*  *   *  +--*  *
   |  |         |  |  |  |   |  |  |  |
   *  *--+--*   *--*  +  *   *  *  +  *
   |        |         |  |   |  |  |  |
   *--*--*--+         *--+   *--*  *--+
   +--*--*      +--*--*      +--*--*
   |     |      |     |      |     |
   *--+  *--*   *--+  *      *--+  *
      |     |      |  |         |  |
      *--+  *   *--*  +--*      *  +--*
         |  |   |        |      |     |
         *--+   *--*--*--+      *--*--+
   +--*--*      +--*  *--*   +--*  *--*
   |     |      |  |  |  |   |  |  |  |
   *  +--*      *  +--*  *   *  +--*  *
   |  |         |        |   |        |
   *  *--+--*   *--*--+  *   *  *--+  *
   |        |         |  |   |  |  |  |
   *--*--*--+         *--+   *--*  *--+
   +--*  *--*   +--*         +--*
   |  |  |  |   |  |         |  |
   *  +  *  *   *  +--*--*   *  +--*--*
   |  |  |  |   |        |   |        |
   *  *--+  *   *--*--+  *   *  *--+  *
   |        |         |  |   |  |  |  |
   *--*--*--+         *--+   *--*  *--+
   +--*         +--*         +--*
   |  |         |  |         |  |
   *  +--*      *  +--*      *  +  *--*
   |     |      |     |      |  |  |  |
   *--*  +--*   *     +--*   *  *--+  *
      |     |   |        |   |        |
      *--*--+   *--*--*--+   *--*--*--+
   +--*
   |  |
   *  +
   |  |
   *  *--+--*
   |        |
   *--*--*--+

Cf. A333455, A333464, A333466, A333795.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333796(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    points = [i + 1 for i in range(n * n) if i % n - i // n == 0]
    for i in points:
        cycles = cycles.including(i)
    return cycles.len()
print([A333796(n) for n in range(2, 10)])

cp's OEIS Frontend

A333513 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of self-avoiding closed paths on an n X k grid which pass through four corners ((0,0), (0,k-1), (n-1,k-1), (n-1,0)).

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A333246 Number of self-avoiding closed paths on an n X n grid which pass through NW corner.

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A333323 Number of self-avoiding closed paths on an n X n grid which pass through NW and SE corners.

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A333247 Number of self-avoiding closed paths on an n X n grid which pass through NW and SW corners.

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A333668 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2*n+2, read by rows, where T(n,k) is the number of 2*(k+2*n-2)-cycles in the n X n grid graph which pass through four corners ((0,0), (0,n-1), (n-1,n-1), (n-1,0)).

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A333759 Number of self-avoiding closed paths in the n X n grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.

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A333795 Number of self-avoiding closed paths on an n X n grid which pass through all points on the two diagonals of the grid.

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A333796 Number of self-avoiding closed paths on an n X n grid which pass through all points on the diagonal connecting NW and SE corners.

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A333668 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2n+2, read by rows, where T(n,k) is the number of 2(k+2*n-2)-cycles in the n X n grid graph which pass through four corners ((0,0), (0,n-1), (n-1,n-1), (n-1,0)).