A014585 -id:A014585 - cp's OEIS frontend

A000532 Number of Hamiltonian paths from NW to SW corners in an n X n grid.

1, 1, 2, 8, 86, 1770, 88418, 8934966, 2087813834, 1013346943033, 1111598871478668, 2568944901392936854, 13251059359839620127088, 145194816279817259193401518, 3524171261632305641165676374930, 182653259988707123426135593460533473

Offset: 1

Russ Cox, Mar 15 1996

Number of walks reaching each cell exactly once.

Oliver R. Bellwood, Table of n, a(n) for n = 1..20 (first 19 terms from KeyTo9(AT)Fans, Ed Wynn)
KeyTo9(AT)Fans, Counting paths in a grid - Chinese web page giving the sequence up to 18 items.
Douglas M. McKenna, Tendril Motifs for Space-Filling, Half-Domino Curves, in: Bridges Finland Conference Proceedings, 2016, pp. 119-126.
Douglas M. McKenna, Are Maximally Unbalanced Hilbert-Style Square-Filling Curve Motifs a Drawing Medium?, Bridges Conf. Proc.; Math., Art, Music, Architecture, Culture (2023) 91-98.

Main diagonal of A271592.
Cf. A181688, A181689, A014524, A014585.
Cf. A001184, A145157, A120443, A003763, A271507, A007764, A121785, A121789.
See also A350148.

More terms from Zhao Hui Du, Jul 08 2008
Edited by Franklin T. Adams-Watters, Jul 03 2009
Name clarified by Andrew Howroyd, Apr 10 2016

A271592 Array read by antidiagonals: T(n,m) = number of directed Hamiltonian walks from NW to SW corners on a grid with n rows and m columns.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 2, 1, 1, 0, 1, 0, 4, 0, 1, 0, 1, 4, 8, 8, 1, 1, 0, 1, 0, 23, 0, 16, 0, 1, 0, 1, 8, 55, 86, 47, 32, 1, 1, 0, 1, 0, 144, 0, 397, 0, 64, 0, 1, 0, 1, 16, 360, 948, 1770, 1584, 264, 128, 1, 1, 0, 1, 0, 921, 0, 11658, 0, 6820, 0, 256, 0, 1

Offset: 1

Andrew Howroyd, Apr 10 2016

			The start of the sequence as table:
* 1 0   0   0     0     0       0       0          0 ...
* 1 1   1   1     1     1       1       1          1 ...
* 1 0   2   0     4     0       8       0         16 ...
* 1 1   4   8    23    55     144     360        921 ...
* 1 0   8   0    86     0     948       0      10444 ...
* 1 1  16  47   397  1770   11658   59946     359962 ...
* 1 0  32   0  1584     0   88418       0    4999752 ...
* 1 1  64 264  6820 52387  909009 8934966  130373192 ...
* 1 0 128   0 28002     0 7503654       0 2087813834 ...
* ...

Andrew Howroyd, Antidiagonals n = 1..27, flattened

Column 4 is aerated A014524, column 5 is A014585.
Rows include A181688, A181689.
Main diagonal is A000532.
Cf. A333580.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A271592(n, k):
    if k == 1: return 1
    universe = tl.grid(k - 1, n - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, n
    paths = GraphSet.paths(start, goal, is_hamilton=True)
    return paths.len()
print([A271592(j + 1, i - j + 1) for i in range(12) for j in range(i + 1)])  # Seiichi Manyama, Mar 28 2020

T(n,m)=0 for n odd and m even, T(1,n)=0 for n>1.

T(2,n)=T(n,1)=T(2*n,2)=1, T(3,2*n+1)=T(n+1,3)=2^n.

A014584 Number of Hamiltonian paths in a 5 X n grid starting at the lower left corner and finishing in the upper right corner.

Original entry on oeis.org

0, 1, 1, 8, 20, 104, 378, 1670, 6706, 28417, 117204, 490865, 2039569, 8512474, 35444636, 147780722, 615715196, 2566325356, 10694300534, 44570089963, 185740837148, 774080813649, 3225945847829, 13444117980220, 56028001091944, 233495908297044, 973089296878098, 4055332929187618, 16900521902518438

Offset: 0

N. J. A. Sloane

nonn

The difference between A014584 and A014585 needs to be clarified. - N. J. A. Sloane, Feb 08 2013

The difference is that this sequence counts Hamiltonian paths that start in the lower left corner and end in the upper right. A014585 counts Hamiltonian paths that start in the lower left and finish in the lower right. - Ruben Zilibowitz, Jul 05 2015

Seiichi Manyama, Table of n, a(n) for n = 0..1000
K. L. Collins and L. B. Krompart, The number of Hamiltonian paths in a rectangular grid, Discrete Math. 169 (1997), 29-38.
Index entries for sequences related to graphs, Hamiltonian

Row n=5 of A333580.

Cf. A014585.

The reference gives a generating function.

Definition clarified by Ruben Zilibowitz, Jul 05 2015

a(24)-a(28) from Seiichi Manyama, Mar 27 2020

A014524 Number of Hamiltonian paths from NW to SW corners in a grid with 2n rows and 4 columns.

Original entry on oeis.org

0, 1, 8, 47, 264, 1480, 8305, 46616, 261663, 1468752, 8244304, 46276385, 259755560, 1458042831, 8184190168, 45938958232, 257861540369, 1447411446840, 8124514782015, 45603992276896, 255981331487648

Offset: 0

N. J. A. Sloane

			Illustration of a(1)=1:
   .__.__.__.
   .__.__.__|
Illustration of a few of the 8 solutions to a(2):
   .__.__.__.    .  .__.__.    .  .__.__.    .__.__.__.
   .__.__.  |    |  |  .__|    |__|  .__|    .__.__.__|
   |__   |  |    |__|  |__.    .__.  |__.    |__.__.__.
   .__|  |__|    .__.__.__|    |  |__.__|    .__.__.__|

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. L. Collins and L. B. Krompart, The number of Hamiltonian paths in a rectangular grid, Discrete Math. 169 (1997), 29-38.
Index entries for sequences related to graphs, Hamiltonian
Index entries for linear recurrences with constant coefficients, signature (7,-9,7,-1).

Even bisection of column 4 of A271592.

Cf. A000532, A181688, A014523, A014585, A003695, A006864.

CoefficientList[Series[x (x + 1)/(x^4 - 7 x^3 + 9 x^2 - 7 x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2013 *)

From Colin Barker, May 20 2013: (Start)
a(n) = 7*a(n-1)-9*a(n-2)+7*a(n-3)-a(n-4).
G.f.: x*(x+1)/(x^4-7*x^3+9*x^2-7*x+1). (End)

Name clarified by Andrew Howroyd, Apr 10 2016

A181689 Number of maximal self-avoiding walks from NW to SW corners of a 5 X n grid.

Original entry on oeis.org

1, 0, 8, 0, 86, 0, 948, 0, 10444, 0, 115056, 0, 1267512, 0, 13963520, 0, 153828832, 0, 1694652176, 0, 18669100976, 0, 205667768400, 0, 2265734756752, 0, 24960420526224, 0, 274975961325264, 0, 3029267044091408, 0, 33371858326057936, 0, 367640393509287824, 0, 4050102862690348880, 0, 44617875206245953552, 0, 491531908055724064720, 0, 5414951194338345409680, 0, 59653698888134291413584, 0, 657173751585588653678864, 0, 7239741169830151881286864

Offset: 1

Sean A. Irvine, Nov 17 2010

All even terms are 0.

Index entries for linear recurrences with constant coefficients, signature (0,11,0,0,0,2).

Row 5 of A271592.

Cf. A000532, A003778, A006865, A014584, A014585, A181688.

I:=[1,0,8,0,86,0]; [n le 6 select I[n] else 11*Self(n-2)+2*Self(n-6): n in [1..50]]; // Wesley Ivan Hurt, Apr 10 2016

A181689:=proc(n) option remember:
if n mod 2 = 0 then 0 elif n=1 then 1 elif n=3 then 8 elif n=5 then 86 else 11*a(n-2)+2*a(n-6); fi; end: seq(A181689(n), n=1..50); # Wesley Ivan Hurt, Apr 10 2016

CoefficientList[Series[(1 - 3*x^2 - 2*x^4)/(1 - 11*x^2 - 2*x^6), {x, 0, 50}], x] (* Wesley Ivan Hurt, Apr 10 2016 *)

x='x+O('x^99); Vec(x*(1-3*x^2-2*x^4)/(1-11*x^2-2*x^6)) \\ Altug Alkan, Apr 11 2016

G.f.: x*(1 - 3*x^2 - 2*x^4)/(1 - 11*x^2 - 2*x^6).

a(n) = 11*a(n-2) + 2*a(n-6) for n>6. - Wesley Ivan Hurt, Apr 10 2016

cp's OEIS Frontend

A000532 Number of Hamiltonian paths from NW to SW corners in an n X n grid.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A271592 Array read by antidiagonals: T(n,m) = number of directed Hamiltonian walks from NW to SW corners on a grid with n rows and m columns.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Formula

A014584 Number of Hamiltonian paths in a 5 X n grid starting at the lower left corner and finishing in the upper right corner.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A014524 Number of Hamiltonian paths from NW to SW corners in a grid with 2n rows and 4 columns.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A181689 Number of maximal self-avoiding walks from NW to SW corners of a 5 X n grid.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula