Search: golygon
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A007219
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Number of golygons of order 8n (or serial isogons of order 8n).
(Formerly M5204)
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+20
6
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1, 28, 2108, 227322, 30276740, 4541771016, 739092675672, 127674038970623, 23085759901610016, 4327973308197103600, 835531767841066680300, 165266721954751746697155, 33364181616540879268092840
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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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.
Eric Weisstein's World of Mathematics, Golygon
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FORMULA
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A006718
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Number of golygons of length 8n.
(Formerly M3707)
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+20
3
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1, 4, 112, 8432, 909288, 121106960, 18167084064, 2956370702688, 510696155882492, 92343039606440064, 17311893232788414400, 3342127071364266721200, 661066887819006986788620, 133456726466163517072371360
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OFFSET
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0,2
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COMMENTS
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A007219 is the main entry for golygons.
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REFERENCES
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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.
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LINKS
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Eric Weisstein's World of Mathematics, Golygon
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FORMULA
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MATHEMATICA
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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 *)
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CROSSREFS
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See A007219 for much more information about golygons.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A101856
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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.
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+10
4
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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
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OFFSET
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1,16
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COMMENTS
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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).
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REFERENCES
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Dudeney, A. K. "An Odd Journey Along Even Roads Leads to Home in Golygon City." Sci. Amer. 263, 118-121, 1990.
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LINKS
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Eric Weisstein's World of Mathematics, Golygon
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EXAMPLE
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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...
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)
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PROG
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(Ruby)
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
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CROSSREFS
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KEYWORD
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nice,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A273089
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Number of lattice n-gons with ordered sides 1, 2, 3, ..., n.
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+10
1
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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
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OFFSET
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1,8
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COMMENTS
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First 16 terms calculated by Stefan Kohl.
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LINKS
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EXAMPLE
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a(8) = 3:
. . . . . ._._._. . . . . . . . . ._._._._._._._.
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. . . . . | . ._| ._._._._._._._. | . . . . . . |
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._._._._._| . | . | . . . . . | | . . . . . . |
| . . . . . . | . | . . . |\. . | | . . . . . . |
| . . . . . . | . | . . . | \ . | | ._._._. . . |
| . . . . . . | . | . . . | .\5 | | | . . | . ./.
| . . . . . . | . | ._._._| . .\| |_| . . | ./5 .
| . . . . . . | . | | . . . . . . . . . . | / . .
|_._._._._._._| . |_| . . . . . . . . . . |/. . .
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a(11) = 5:
. . . . . . . . . . . ._._._._._._._._._. . . . . . . .
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. . . . . . . ._._._. . . . .\. . . . . | . . . . . . .
. . . . . . . | . . | . . . . .10 . . . | . . . . . . .
. . . . . . ._| . . | . . . . . . \ . . | . . . . . . .
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. . . . . . | . . . . \ . . . . . . . | |_._._._._._._.
|\. . . . . | . . . . . 5 \ . . . . . | . . . . . . . |
| \ . . . . | . ._._._._._._. . . . . | . . . . . . . |
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| . . 10. . | . | . . . . . . . . . . | . . . | . 5 . |
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| . . . . \ | . | . . . . . . . . . . | ._._._| . . . .
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|_._._._._._._._| . . . . . . . . . . |_| . . . . . . .
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. ._._._. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. | . . | . . . . . . . . . . . . . . . . . ./| . . . . . . . .
._| . . | . . . . . . . . . . . . . . . . ./. | . . . . . . . .
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| . . . .\. . . . . . . . . . . . . . .10 . . | . . . . . . . .
| . . . . . 5 . . . . . . . . . . . . / . . . | . . . . . . . .
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| . . . . . . . | . . . . . . . . | . . . . . |_._._._._._._._.
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| . . . . . . . |_._._._._._._. . | . . . . . . . . . . . . . |
.\. . . . . . . . . . . . . . | . | . . . | \ . . . . . . . . |
. \ . . . . . . . . . . . . . | . | . . . | . .5. . . . . . . |
. . \ . . . . . . . . . . . . | . | . . . | . . .\._._._._._._|
. . . \ . . . . . . . . . . . | . | ._._._| . . . . . . . . . .
. . . 10. . . . . . . . . . . | . | | . . . . . . . . . . . . .
. . . . \ . . . . . . . . . . | . |_| . . . . . . . . . . . . .
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| . . 10. . . . . . . . . . . .
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|_| . . | . . ./| . . . . . . |
. . . . | . 5 . | . . . . . . |
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. . . . . . . . |_._._._._._._|
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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Typo in a(15) corrected and a(17)-a(27) added by Giovanni Resta, Mar 26 2020
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STATUS
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approved
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