A333513 -id:A333513 - cp's OEIS frontend

A333466 Number of self-avoiding closed paths on an n X n grid which pass through four corners ((0,0), (0,n-1), (n-1,n-1), (n-1,0)).

1, 1, 11, 373, 44930, 17720400, 22013629316, 84579095455492

Offset: 2

Seiichi Manyama, Mar 22 2020

a(11) = 36061721109572407840288. - Seiichi Manyama, Apr 07 2020

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

Main diagonal of A333513.

Cf. A003763, A140517, A333246, A333247, A333323.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333466(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()
print([A333466(n) for n in range(2, 10)])

def search(x, y, n, used)
  return 0 if x < 0 || n <= x || y < 0 || n <= y || used[x + y * n]
  return 1 if x == 0 && y == 1 && [n - 1, n * (n - 1), n * n - 1].all?{|i| used[i] == true}
  cnt = 0
  used[x + y * n] = true
  @move.each{|mo|
    cnt += search(x + mo[0], y + mo[1], n, used)
  }
  used[x + y * n] = false
  cnt
end
def A(n)
  return 1 if n < 3
  @move = [[1, 0], [-1, 0], [0, 1], [0, -1]]
  used = Array.new(n * n, false)
  search(0, 0, n, used)
end
def A333466(n)
  (2..n).map{|i| A(i)}
end
p A333466(6)

A333758 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of self-avoiding closed paths in the n X k grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 11, 5, 1, 1, 11, 36, 36, 11, 1, 1, 21, 122, 191, 122, 21, 1, 1, 43, 408, 1123, 1123, 408, 43, 1, 1, 85, 1371, 6410, 11346, 6410, 1371, 85, 1, 1, 171, 4599, 37165, 113748, 113748, 37165, 4599, 171, 1

Offset: 2

Seiichi Manyama, Apr 04 2020

			T(4,3) = 3;
   +--+--+   +--+--+   +--+--+
   |     |   |     |   |     |
   +--*  +   +  *--+   +     +
      |  |   |  |      |     |
   +--*  +   +  *--+   +     +
   |     |   |     |   |     |
   +--+--+   +--+--+   +--+--+
Square array T(n,k) begins:
  1,  1,   1,    1,      1,       1,        1, ...
  1,  1,   3,    5,     11,      21,       43, ...
  1,  3,  11,   36,    122,     408,     1371, ...
  1,  5,  36,  191,   1123,    6410,    37165, ...
  1, 11, 122, 1123,  11346,  113748,  1153742, ...
  1, 21, 408, 6410, 113748, 2002405, 35669433, ...

Rows n=2..7 give: A000012, A001045(n-1), A333760, A358696, A358697, A358698.
Main diagonal gives A333759.
Cf. A333513.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333758(n, k):
    universe = tl.grid(n - 1, k - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    points = [i for i in range(1, k * n + 1) if i % k < 2 or ((i - 1) // k + 1) % n < 2]
    for i in points:
        cycles = cycles.including(i)
    return cycles.len()
print([A333758(j + 2, i - j + 2) for i in range(11 - 1) for j in range(i + 1)])

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

A333515 Number of self-avoiding closed paths on an n X 5 grid which pass through four corners ((0,0), (0,4), (n-1,4), (n-1,0)).

Original entry on oeis.org

1, 7, 49, 373, 3105, 26515, 227441, 1953099, 16782957, 144262743, 1240194297, 10662034451, 91663230249, 788046822891, 6775004473757, 58246174168047, 500755017859261, 4305100014182879, 37011883913816129, 318199242452585915, 2735628331213604009, 23518793814422304163

Offset: 2

Seiichi Manyama, Mar 25 2020

nonn

Also number of self-avoiding closed paths on a 5 X n grid which pass through four corners ((0,0), (0,n-1), (4,n-1), (4,0)).

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

Seiichi Manyama, Table of n, a(n) for n = 2..1000

Column k=5 of A333513.

# 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()
def A333515(n):
    return A333513(n, 5)
print([A333515(n) for n in range(2, 25)])

Conjectures from Chai Wah Wu, Jan 17 2024: (Start)
a(n) = 13*a(n-1) - 45*a(n-2) + 66*a(n-3) - 17*a(n-4) - 209*a(n-5) + 151*a(n-6) + 140*a(n-7) - 112*a(n-8) - 48*a(n-9) + 50*a(n-10) + 28*a(n-11) for n > 12.
G.f.: x^2*(4*x^7 + 2*x^6 - 29*x^5 - 16*x^4 + 15*x^3 - 3*x^2 + 6*x - 1)/(28*x^11 + 50*x^10 - 48*x^9 - 112*x^8 + 140*x^7 + 151*x^6 - 209*x^5 - 17*x^4 + 66*x^3 - 45*x^2 + 13*x - 1). (End)

A333514 Number of self-avoiding closed paths on an n X 4 grid which pass through four corners ((0,0), (0,3), (n-1,3), (n-1,0)).

Original entry on oeis.org

1, 3, 11, 49, 229, 1081, 5123, 24323, 115567, 549253, 2610697, 12409597, 58988239, 280398495, 1332867179, 6335755801, 30116890013, 143160058769, 680508623307, 3234784886251, 15376488953815, 73091850448509, 347440733910081, 1651552982759797, 7850625988903223

Offset: 2

Seiichi Manyama, Mar 25 2020

nonn

Also number of self-avoiding closed paths on a 4 X n grid which pass through four corners ((0,0), (0,n-1), (3,n-1), (3,0)).

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

Index entries for linear recurrences with constant coefficients, signature (7,-12,7,-3,-2).

Column k=4 of A333513.

N=40; x='x+O('x^N); Vec(x^2*(1-4*x+2*x^2+x^3)/(1-7*x+12*x^2-7*x^3+3*x^4+2*x^5))

# 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()
def A333514(n):
    return A333513(4, n)
print([A333514(n) for n in range(2, 15)])

G.f.: x^2*(1-4*x+2*x^2+x^3)/(1-7*x+12*x^2-7*x^3+3*x^4+2*x^5).

a(n) = 7*a(n-1) - 12*a(n-2) + 7*a(n-3) - 3*a(n-4) - 2*a(n-5) for n > 6.

A358713 Number of self-avoiding closed paths on an n X 7 grid which pass through four corners ((0,0), (0,6), (n-1,6), (n-1,0)).

Original entry on oeis.org

1, 41, 1081, 26515, 674292, 17720400, 471468756, 12570253556, 335101401877, 8932110760401, 238088717357193, 6346541968642151, 169176879483125528, 4509681115981777876, 120212775466066851264, 3204464007623702644476, 85420126381414152110475, 2277010595175522782497635

Offset: 2

Seiichi Manyama, Nov 28 2022

nonn

Also number of self-avoiding closed paths on a 7 X n grid which pass through four corners ((0,0), (0,6), (n-1,6), (n-1,0)).

Seiichi Manyama, Table of n, a(n) for n = 2..50

Column k=7 of A333513.

Cf. A358698.

A358712 Number of self-avoiding closed paths on an n X 6 grid which pass through four corners ((0,0), (0,5), (n-1,5), (n-1,0)).

Original entry on oeis.org

1, 17, 229, 3105, 44930, 674292, 10217420, 154980130, 2350703747, 35658264301, 540957030465, 8206939419403

Offset: 2

Seiichi Manyama, Nov 28 2022

Also number of self-avoiding closed paths on a 6 X n grid which pass through four corners ((0,0), (0,5), (n-1,5), (n-1,0)).

Column k=6 of A333513.

Cf. A358697.

cp's OEIS Frontend

A333466 Number of self-avoiding closed paths on an n X n grid which pass through four corners ((0,0), (0,n-1), (n-1,n-1), (n-1,0)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Ruby

A333758 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of self-avoiding closed paths in the n X k grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

Formula

A333515 Number of self-avoiding closed paths on an n X 5 grid which pass through four corners ((0,0), (0,4), (n-1,4), (n-1,0)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A333514 Number of self-avoiding closed paths on an n X 4 grid which pass through four corners ((0,0), (0,3), (n-1,3), (n-1,0)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

A358713 Number of self-avoiding closed paths on an n X 7 grid which pass through four corners ((0,0), (0,6), (n-1,6), (n-1,0)).

Views

Author

Keywords

Comments

Links

Crossrefs

A358712 Number of self-avoiding closed paths on an n X 6 grid which pass through four corners ((0,0), (0,5), (n-1,5), (n-1,0)).

Views

Author

Keywords

Comments

Crossrefs