cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A078602 Number of ways to lace a shoe that has n pairs of eyelets.

Original entry on oeis.org

1, 2, 21, 601, 34278, 3144357, 421928841, 77832868334
Offset: 1

Views

Author

N. J. A. Sloane, Dec 11 2002

Keywords

Comments

The lace must pass through each eyelet exactly once, must begin and end at the extreme pair of eyelets and cannot pass in order though three adjacent eyelets that are in a line.
The lace is "directed": reversing the order of eyelets along the path counts as a different solution.

Examples

			a(3) = 21: label the eyelets 1,2,3 from front to back on the left side then 4,5,6 from back to front on the right side. The lacings are: 124356 154326 153426 142536 145236 132546 135246 together with the following lacings and their mirror images: 125346 124536 125436 152346 153246 152436 154236.

Links

Crossrefs

Cf. A078675 (undirected solutions), A078676 (symmetric solutions). See A078601 and A078629 for other ways of counting lacings.
Cf. A072503.

Extensions

a(7) and a(8) from Hugo Pfoertner, Jan 22 2005

A078601 Number of ways to lace a shoe that has n pairs of eyelets, assuming the lacing satisfies certain conditions.

Original entry on oeis.org

1, 3, 42, 1080, 51840, 3758400, 382838400, 52733721600, 9400624128000, 2105593491456000, 579255485276160000, 191957359005941760000, 75420399121328701440000, 34668462695110852608000000, 18432051070888873171353600000, 11223248177765618214764544000000, 7759395812038133743242706944000000
Offset: 1

Views

Author

N. J. A. Sloane, Dec 11 2002

Keywords

Comments

The lace must follow a Hamiltonian path through the 2n eyelets. At least one of the neighbors of every eyelet must be on the other side of the shoe.
The lace is "undirected": reversing the order of eyelets along the path does not count as a different solution.

Examples

			Label the eyelets 1, ..., n from front to back on the left and from n+1, ..., 2n from back to front on the right. For n=2 the three solutions are 1 2 3 4, 3 1 2 4, 1 3 2 4.
For n=3 the first few solutions are 2 4 1 3 5 6, 1 4 2 3 5 6, 2 1 4 3 5 6, 1 2 4 3 5 6, 1 3 4 2 5 6, 3 1 4 2 5 6, 1 4 3 2 5 6, 3 4 1 2 5 6, 3 4 2 1 5 6, 2 4 3 1 5 6, 3 2 4 1 5 6, 2 3 4 1 5 6, 2 3 5 1 4 6, 3 2 5 1 4 6, 2 5 3 1 4 6, 3 5 2 1 4 6, ...

Links

Crossrefs

See A078602 and A078629 for other ways of counting lacings.
Cf. A123385.

Programs

  • Maple
    A078601 := n->((n!)^2/2)*add(binomial(n-k,k)^2/(n-k),k=0..floor(n/2));
  • Mathematica
    a[n_] := If[n == 1, 1, n!^2/2 Sum[Binomial[n-k, k]^2/(n-k), {k, 0, n/2}]];
a /@ Range[1, 17] (* Jean-François Alcover, Oct 01 2019 *)
  • PARI
    a(n)=if(n>1,n!^2*sum(k=0,n\2,binomial(n-k, k)^2/(n-k))/2,1) \\ Charles R Greathouse IV, Sep 10 2015
  • Python
    from sympy import factorial, binomial
a = lambda n:((factorial(n)**2)>>1) * sum((binomial(n-k,k)**2)/(n-k) for k in range(0,(n>>1)+1)) if n > 1 else 1
print([a(n) for n in range(1, 18)]) # Darío Clavijo, Mar 06 2024

Formula

a(1)=1; for n > 1, a(n) = ((n!)^2/2)*Sum_{k=0..floor(n/2)} binomial(n-k, k)^2/(n-k).

A078674 Number of ways to lace a shoe that has n pairs of eyelets, assuming the lacing satisfies certain conditions.

Original entry on oeis.org

1, 3, 50, 2122, 155712, 17441962
Offset: 1

Views

Author

N. J. A. Sloane, Dec 16 2002

Keywords

Comments

The lace must follow a Hamiltonian path through the 2n eyelets and cannot pass in order though three adjacent eyelets that are in a line.
The lace is "undirected": reversing the order of eyelets along the path does not count as a different solution (cf. A078629).

Links

Crossrefs

Apart from initial term, A078629/2.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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