A007219
Number of golygons of order 8n (or serial isogons of order 8n).
Original entry on oeis.org
1, 28, 2108, 227322, 30276740, 4541771016, 739092675672, 127674038970623, 23085759901610016, 4327973308197103600, 835531767841066680300, 165266721954751746697155, 33364181616540879268092840
Offset: 1
- 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.
- 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
-
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 *)
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.
Original entry on oeis.org
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
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)
- Dudeney, A. K. "An Odd Journey Along Even Roads Leads to Home in Golygon City." Sci. Amer. 263, 118-121, 1990.
- 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
-
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
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