A007219 Number of golygons of order 8n (or serial isogons of order 8n). (Formerly M5204)

1, 28, 2108, 227322, 30276740, 4541771016, 739092675672, 127674038970623, 23085759901610016, 4327973308197103600, 835531767841066680300, 165266721954751746697155, 33364181616540879268092840

Offset

1,2

Comments

A golygon of order N is a closed path along the streets of the Manhattan grid with successive edge lengths of 1,2,3,...,N (returning to the starting point after the edge of length N), and which makes a 90-degree turn (left or right) after each edge.

It is known that the order N must be a multiple of 8.

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 92.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..100

A. K. Dewdney, An odd journey along even roads leads to home in Golygon City, Mathematical Recreations Column, Scientific American, July 1990, pp. 118-121.

A. K. Dewdney, Illustration of the unique golygon of order 8, from the article "An odd journey along even roads leads to home in Golygon City", Mathematical Recreations Column, Scientific American, July 1990, pp. 118-121.

A. K. Dewdney, Illustration of the 28 golygons of order 16, from the article "An odd journey along even roads leads to home in Golygon City", Mathematical Recreations Column, Scientific American, July 1990, pp. 118-121.

Adam P. Goucher, Golygons and golyhedra

L. Sallows, M. Gardner, R. K. Guy and D. E. Knuth, Serial isogons of 90 degrees, Math. Mag. 64 (1991), 315-324.

Eric Weisstein's World of Mathematics, Golygon

Formula

a(n) = A006718(n)/4. - Charles R Greathouse IV, Apr 29 2012

a(n) ~ 3*2^(8*n-6)/(Pi*n^2*(4*n+1)). - Vaclav Kotesovec, Dec 09 2013

Mathematica

p1[n_] := Product[x^k + 1, {k, 1, n - 1, 2}] // Expand; p2[n_] := Product[x^k + 1, {k, 1, n/2}] // Expand; c[n_] := Coefficient[p1[n], x, n^2/8] * Coefficient[p2[n], x, n (n/2 + 1)/8]; a[n_] := c[8*n]/4; Table[a[n], {n, 1, 13}] (* Jean-François Alcover, Jul 24 2013, after Eric W. Weisstein *)

Crossrefs

Cf. A060005, A107350.

See also A006718.

Keyword

nonn,easy,nice

Author

Simon Plouffe

Extensions

Two more terms from N. J. A. Sloane (from the reference), May 23 2005

Status

approved

A006718 Number of golygons of length 8n. (Formerly M3707)

1, 4, 112, 8432, 909288, 121106960, 18167084064, 2956370702688, 510696155882492, 92343039606440064, 17311893232788414400, 3342127071364266721200, 661066887819006986788620, 133456726466163517072371360

Offset

0,2

Comments

A007219 is the main entry for golygons.

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 92.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..200

Eric Weisstein's World of Mathematics, Golygon

Formula

a(n) = 4 * A007219(n) for n > 0. - Charles R Greathouse IV, Apr 29 2012

a(n) = A060468(n) * A292476(2*n) = A063865(4*n) * A292476(2*n). - Seiichi Manyama, Sep 18 2017

Mathematica

p1[n_] := Product[x^k + 1, {k, 1, n - 1, 2}] // Expand; p2[n_] := Product[x^k + 1, {k, 1, n/2}] // Expand; c[n_] := Coefficient[p1[n], x, n^2/8] * Coefficient[p2[n], x, n (n/2 + 1)/8]; a[n_] := c[8*n]; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Jul 24 2013, after Eric W. Weisstein *)

Crossrefs

Cf. A060468, A063865, A292476.

See A007219 for much more information about golygons.

Keyword

nonn

Author

N. J. A. Sloane

Extensions

a(0) = 1 prepended by Seiichi Manyama, Sep 18 2017

Status

approved

A101856 Number of non-intersecting polygons that it is possible for an accelerating ant to produce with n steps (rotations & reflections not included). On step 1 the ant moves forward 1 unit, then turns left or right and proceeds 2 units, then turns left or right until at the end of its n-th step it arrives back at its starting place.

0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 25, 67, 0, 0, 0, 0, 0, 0, 515, 1259, 0, 0, 0, 0, 0, 0, 15072, 41381, 0, 0, 0, 0, 0, 0, 588066, 1651922, 0, 0, 0, 0, 0, 0, 25263990, 73095122, 0, 0, 0, 0, 0, 0, 1194909691, 3492674650, 0, 0, 0, 0, 0, 0

Offset

1,16

Comments

This walk by an accelerating ant can only arrive back at the starting point after n steps where n is 0 or -1 mod(8).

References

Dudeney, A. K. "An Odd Journey Along Even Roads Leads to Home in Golygon City." Sci. Amer. 263, 118-121, 1990.

Links

Table of n, a(n) for n=1..70.

Bert Dobbelaere, C++ program

L. C. F. Sallows, New Pathways in Serial Isogons, Math. Intell. 14, 55-67, 1992.

Lee Sallows, Martin Gardner, Richard K. Guy and Donald Knuth, Serial Isogons of 90 Degrees, Math Mag. 64, 315-324, 1991.

Eric Weisstein's World of Mathematics, Golygon

Example

For example: a(7) = 1 because of the following solution:
655555...
6....4...
6....4...
6....4...
6....4333
6.......2
777777712
where the ant starts at the "1" and moves right 1 space, up 2 spaces and so on...
From Seiichi Manyama, Sep 23 2017: (Start)
a(8) = 1 because of the following solution:
(0, 0) -> (1, 0) -> (1, 2) -> (-2, 2) -> (-2, -2) -> (-7, -2) -> (-7, -8) -> (0, -8) -> (0, 0).
.....4333
.....4..2
.....4.12
.....4.8.
655555.8.
6......8.
6......8.
6......8.
6......8.
6......8.
77777778.
a(15) = 1 because of the following solution:
(0, 0) -> (1, 0) -> (1, 2) -> (4, 2) -> (4, -2) -> (-1, -2) -> (-1, -8) -> (-8, -8) -> (-8, -16) -> (-17, -16) -> (-17, -26) -> (-28, -26) -> (-28, -14) -> (-15, -14) -> (-15, 0) -> (0, 0).
a(16) = 3 because of the following solutions:
(0, 0) -> (1, 0) -> (1, 2) -> (4, 2) -> (4, 6) -> (-1, 6) -> (-1, 12) -> (-8, 12) -> (-8, 20) -> (-17, 20) -> (-17, 10) -> (-28, 10) -> (-28, -2) -> (-15, -2) -> (-15, -16) -> (0, -16) -> (0, 0),
(0, 0) -> (1, 0) -> (1, 2) -> (4, 2) -> (4, 6) -> (-1, 6) -> (-1, 0) -> (-8, 0) -> (-8, -8) -> (-17, -8) -> (-17, -18) -> (-28, -18) -> (-28, -30) -> (-15, -30) -> (-15, -16) -> (0, -16) -> (0, 0),
(0, 0) -> (1, 0) -> (1, 2) -> (4, 2) -> (4, -2) -> (-1, -2) -> (-1, -8) -> (-8, -8) -> (-8, 0) -> (-17, 0) -> (-17, -10) -> (-28, -10) -> (-28, 2) -> (-15, 2) -> (-15, 16) -> (0, 16) -> (0, 0). (End)

Prog

(Ruby)
def A101856(n)
  ary = [0, 0]
  b_ary = [[[0, 0], [1, 0], [1, 1], [1, 2]]]
  s = 4
  (3..n).each{|i|
    s += i
    t = 0
    f_ary, b_ary = b_ary, []
    if i % 2 == 1
      f_ary.each{|a|
        b = a.clone
        x, y = *b[-1]
        b += (1..i).map{|j| [x + j, y]}
        b_ary << b if b.uniq.size == s
        t += 1 if b[-1] == [0, 0] && b.uniq.size == s - 1
        c = a.clone
        x, y = *c[-1]
        c += (1..i).map{|j| [x - j, y]}
        b_ary << c if c.uniq.size == s
        t += 1 if c[-1] == [0, 0] && c.uniq.size == s - 1
      }
    else
      f_ary.each{|a|
        b = a.clone
        x, y = *b[-1]
        b += (1..i).map{|j| [x, y + j]}
        b_ary << b if b.uniq.size == s
        t += 1 if b[-1] == [0, 0] && b.uniq.size == s - 1
        c = a.clone
        x, y = *c[-1]
        c += (1..i).map{|j| [x, y - j]}
        b_ary << c if c.uniq.size == s
        t += 1 if c[-1] == [0, 0] && c.uniq.size == s - 1
      }
    end
    ary << t
  }
  ary[0..n - 1]
end
p A101856(16) # Seiichi Manyama, Sep 24 2017

Crossrefs

Cf. A101857, A006718, A292793.

Keyword

nice,nonn

Author

Gordon Hamilton, Jan 27 2005

Extensions

a(31)-a(70) from Bert Dobbelaere, Jan 01 2019

Status

approved

A273089 Number of lattice n-gons with ordered sides 1, 2, 3, ..., n.

0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 5, 6, 0, 0, 584, 882, 0, 0, 18026, 194741, 0, 0, 644414, 960834, 0, 0, 229910636

Offset

1,8

Comments

First 16 terms calculated by Stefan Kohl.

Links

Table of n, a(n) for n=1..27.

Stefan Kohl, Lattice n-gons with ordered side lengths 1,2,3,…,n, Answer on MathOverflow, May 4, 2016.

MathOverflow, Lattice n-gons with ordered side lengths 1,2,3,...,n

Giovanni Resta, Examples of n-gons with minimal and maximal area

L. Sallows, M. Gardner, R. K. Guy and D. E. Knuth, Serial isogons of 90 degrees, Math. Mag. 64 (1991), 315-324.

Wikipedia, Golygon

Example

a(8) = 3:
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.
a(11) = 5:
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.
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  . . . . \ . . . . . . . . . . | . |_| . . . . . . . . . . . . .
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.
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  | | . . | . . . . . . . . . . |
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  . . . . | . 5 . | . . . . . . |
  . . . . |/. . . | . . . . . . |
  . . . . . . . . | . . . . . . |
  . . . . . . . . | . . . . . . |
  . . . . . . . . |_._._._._._._|
. - Hugo Pfoertner, Mar 20 2020

Crossrefs

Cf. A007219.

Keyword

nonn,hard,more

Author

Bernardo Recamán, May 14 2016

Extensions

Typo in a(15) corrected and a(17)-a(27) added by Giovanni Resta, Mar 26 2020

Status

approved