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.

Sequence in context: A202811 A285749 A276702 * A203751 A178187 A184134

Adjacent sequences: A007216 A007217 A007218 * A007220 A007221 A007222

Keyword

nonn,easy,nice

Author

Simon Plouffe

Extensions

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

Status

approved