login
Search: golygon
     Sort: relevance | references | number | modified | created      Format: long | short | data
Number of golygons of order 8n (or serial isogons of order 8n).
(Formerly M5204)
+20
6
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
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
See also A006718.
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Two more terms from N. J. A. Sloane (from the reference), May 23 2005
STATUS
approved
Number of golygons of length 8n.
(Formerly M3707)
+20
3
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
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
See A007219 for much more information about golygons.
KEYWORD
nonn
EXTENSIONS
a(0) = 1 prepended by Seiichi Manyama, Sep 18 2017
STATUS
approved
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.
+10
4
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
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
KEYWORD
nice,nonn
AUTHOR
Gordon Hamilton, Jan 27 2005
EXTENSIONS
a(31)-a(70) from Bert Dobbelaere, Jan 01 2019
STATUS
approved
Number of lattice n-gons with ordered sides 1, 2, 3, ..., n.
+10
1
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
Stefan Kohl, Lattice n-gons with ordered side lengths 1,2,3,…,n, Answer on MathOverflow, May 4, 2016.
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:
. . . . . ._._._. . . . . . . . . ._._._._._._._.
. . . . . | . . | . . . . . . . . | . . . . . . |
. . . . . | . ._| ._._._._._._._. | . . . . . . |
. . . . . | . | . | . . . . . . | | . . . . . . |
._._._._._| . | . | . . . . . | | . . . . . . |
| . . . . . . | . | . . . |\. . | | . . . . . . |
| . . . . . . | . | . . . | \ . | | ._._._. . . |
| . . . . . . | . | . . . | .\5 | | | . . | . ./.
| . . . . . . | . | ._._._| . .\| |_| . . | ./5 .
| . . . . . . | . | | . . . . . . . . . . | / . .
|_._._._._._._| . |_| . . . . . . . . . . |/. . .
.
a(11) = 5:
. . . . . . . . . . . ._._._._._._._._._. . . . . . . .
. . . . . . . . . . . .\. . . . . . . . | . . . . . . .
. . . . . . . . . . . . . \ . . . . . . | . . . . . . .
. . . . . . . ._._._. . . . .\. . . . . | . . . . . . .
. . . . . . . | . . | . . . . .10 . . . | . . . . . . .
. . . . . . ._| . . | . . . . . . \ . . | . . . . . . .
. . . . . . | . . . | . . . . . . . .\. | . . . . . . .
. . . . . . | . . . | . . . . . . . . | | . . . . . . .
. . . . . . | . . . . \ . . . . . . . | |_._._._._._._.
|\. . . . . | . . . . . 5 \ . . . . . | . . . . . . . |
| \ . . . . | . ._._._._._._. . . . . | . . . . . . . |
| .\. . . . | . | . . . . . . . . . . | . . . . . . . |
| . .\. . . | . | . . . . . . . . . . | . . . | \ . . |
| . . 10. . | . | . . . . . . . . . . | . . . | . 5 . |
| . . . \ . | . | . . . . . . . . . . | . . . | . . \ |
| . . . . \ | . | . . . . . . . . . . | ._._._| . . . .
| . . . . .\| . | . . . . . . . . . . | | . . . . . . .
|_._._._._._._._| . . . . . . . . . . |_| . . . . . . .
.
. ._._._. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. | . . | . . . . . . . . . . . . . . . . . ./| . . . . . . . .
._| . . | . . . . . . . . . . . . . . . . ./. | . . . . . . . .
| . . . | . . . . . . . . . . . . . . . . / . | . . . . . . . .
| . . . | . . . . . . . . . . . . . . . ./. . | . . . . . . . .
| . . . .\. . . . . . . . . . . . . . .10 . . | . . . . . . . .
| . . . . . 5 . . . . . . . . . . . . / . . . | . . . . . . . .
| . . . . . . .\. . . . . . . . . . / . . . . | . . . . . . . .
| . . . . . . . | . . . . . . . . ./. . . . . | . . . . . . . .
| . . . . . . . | . . . . . . . . | . . . . . |_._._._._._._._.
| . . . . . . . | . . . . . . . . | . . . . . . . . . . . . . |
| . . . . . . . | . . . . . . . . | . . . . . . . . . . . . . |
| . . . . . . . | . . . . . . . . | . . . . . . . . . . . . . |
| . . . . . . . |_._._._._._._. . | . . . . . . . . . . . . . |
.\. . . . . . . . . . . . . . | . | . . . | \ . . . . . . . . |
. \ . . . . . . . . . . . . . | . | . . . | . .5. . . . . . . |
. . \ . . . . . . . . . . . . | . | . . . | . . .\._._._._._._|
. . . \ . . . . . . . . . . . | . | ._._._| . . . . . . . . . .
. . . 10. . . . . . . . . . . | . | | . . . . . . . . . . . . .
. . . . \ . . . . . . . . . . | . |_| . . . . . . . . . . . . .
. . . . . \ . . . . . . . . . |
. . . . . .\._._._._._._._._._|
.
. . . . . . . . . . . . . . . .
|\. . . . . . . . . . . . . . .
| .\. . . . . . . . . . . . . .
| . \ . . . . . . . . . . . . .
| . . \ . . . . . . . . . . . .
| . . 10. . . . . . . . . . . .
| . . . .\. . . . . . . . . . .
| . . . . \ . . . . . . . . . .
| . . . . . \_._._._._._._._._.
| ._._._. . . . . . . . . . . |
| | . . | . . . . . . . . . . |
|_| . . | . . ./| . . . . . . |
. . . . | . 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

Search completed in 0.005 seconds