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

A078700 Number of symmetric ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.

Original entry on oeis.org

1, 2, 6, 30, 192, 1560, 15120, 171360, 2217600, 32296320, 522547200, 9300614400, 180583603200, 3798482688000, 86044973414400, 2088355965696000, 54064489070592000, 1487129136869376000, 43312058119249920000
Offset: 1

Views

Author

Hugo Pfoertner, Dec 18 2002

Keywords

Comments

The lace must pass through each eyelet exactly one and must begin and end at the extreme pair of eyelets.

Examples

			a(3) = 6: 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 symmetric lacings are: 124356 154326 153426 142536 145236 135246.

Links

Crossrefs

Cf. A000045, A005922, A014417, A078676, A078698, A078702.

Programs

  • Maple
    ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card <= 1)}, labelled]: seq(combstruct[count](ZL, size=n), n=0..18); # Zerinvary Lajos, Mar 26 2008
  • Mathematica
    Table[Fibonacci[n + 1]*(n - 1)!, {n, 1, 19}] (* Zerinvary Lajos, Jul 09 2009 *)
  • Scheme
    (define (A078700 n) (* (A000142 (- n 1)) (A000045 (+ n 1)))) ;; Antti Karttunen, Jan 02 2007

Formula

a(n) = (n-1)!*Fibonacci(n+1) = A000142(n-1)*A000045(n+1). - Conjectured by Vladeta Jovovic, Jan 23 2005, proved by Antti Karttunen, Jan 06 2007.
Proof of Jovovic's conjecture from Antti Karttunen, Jan 06 2007: (Start)
Because of the symmetry and the beginning and ending conditions, we need to consider only n-1 eyelets on the other side. If considering only the distance from the starting and ending eyelets (the "level" of each eyelet-pair) through which the lace is traveling (but ignoring on which side it is), the lace will induce some permutation of {1..(n-1)} after it has visited half of the eyelets (and the remaining half of its route is wholly determined by the symmetry). This gives the factor (n-1)!. Independently of this, the condition that each eyelet has at least one direct connection to the opposite side, means that there is a simple bijection with binary strings of length n with no two consecutive 0's. I.e. we mark 0 when the lace stays on the same side and 1 when it crosses to the other side.
From the starting eyelet the lace can either cross to the other side (but not to the top one) or stay on the same side. However, after visiting half of the eylets, the lace MUST cross to the other side (on the same level), so this leaves n-1 eyelets where the choice is free, except that there can be no two consecutive 0's on the route. This gives the factor Fibonacci(n+1). (End)

Extensions

More terms added by Antti Karttunen, Jan 02 2007

A078675 Number of ways to lace a shoe that has n pairs of eyelets.

Original entry on oeis.org

1, 2, 14, 322, 17314, 1573952, 210985926, 38916737688
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 "undirected": reversing the order of eyelets along the path does not count as a different solution.

Examples

			a(3) = 14: 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 125346 124536 125436 152346 153246 152436 154236.

Links

Crossrefs

Cf. A078602 for directed solutions, A078676 for symmetric solutions.
a(n) = ( A078602(n) + A078676(n) ) / 2

Extensions

a(7) and a(8) from Hugo Pfoertner, Jan 22 2005
Showing 1-3 of 3 results.

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

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