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A141292
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Conjectured values for minimal number of pieces required in a 2n-gon to square dissection that uses translation alone.
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0
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1, 5, 9, 12, 15, 19, 22, 25, 28, 31, 35, 38, 42, 47, 50, 53, 56, 60, 63, 67, 72, 76, 79, 82, 85, 89, 93, 100, 103, 106, 109, 113, 117, 121, 126, 130, 133, 136, 139, 143, 147, 156, 160, 163, 166, 169, 174, 177, 182, 186, 189, 192, 196, 202, 205, 214, 217, 220, 223
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OFFSET
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2,2
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REFERENCES
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Boltyanskii, V.G., Equivalent and Equidecomposable Figures, D.C. Heath and Company, Boston, 1963.
G. N. Frederickson, Dissections Plane and Fancy, Camb. 1997.
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LINKS
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EXAMPLE
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a(2)=1 because a regular 4-gon-to-square dissection can be accomplished with a single "piece". Busschop gave a 5-piece hexagon-to-square dissection using translations alone,so a(3)=5 (see Frederickson, p. 118). Further terms in the sequence are obtained by a systematic process for cutting the original 2n-gon, and the algorithm for generating these terms is given below. - Pamela Pierce (ppierce(AT)wooster.edu), Sep 03 2009
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MATHEMATICA
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b1 := (n, k) -> 2*sin(Pi*(2*k-1)/n)
b2 := (n, k) -> 2*sin(Pi*(2*k+1)/n)
w1 := (n, k) -> b1(n, k)+b2(n, k)
w2 := (n, k) -> sqrt((1/2)*n*sin(2*Pi/n))
h1 := (n, k) -> cos(Pi*(2*k-1)/n)-cos(Pi*(2*k+1)/n)
h2 := (n, k) -> w1(n, k)*h1(n, k)/w2(n, k)
a := (n, k) -> floor(w2(n, k)/w1(n, k))*h2(n, k)/h1(n, k)
kp := (n, k) -> 3*signum(w1(n, k)-w2(n, k))+3+((1/2)*signum(w2(n, k)-w1(n, k))+1/2)* (3*floor(w2(n, k)/w1(n, k))+9/2+(1/2)*signum(w2(n, k)-floor(w2(n, k)/w1(n, k))*w1(n, k)-(1/2-(1/2)*a(n, k))*w1(n, k)-a(n, k)*b2(n, k)))
P := n-> 9/2+sum(kp(n, k), k = 1 .. floor((1/4)*n)-1)+1.5*signum((1/4)*n-floor((1/4)*n)-.25)
[seq([2*i, P(2*i)], i = 3 .. 100)]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Pamela Pierce (PPierce(AT)wooster.edu), Jeffrey Willert (jwillert09(AT)wooster.edu) and Wenyuan Wu (wwu11(AT)wooster.edu), Aug 01 2008, Aug 12 2008
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EXTENSIONS
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Entry revised by Pamela Pierce (PPierce(AT)wooster.edu), John Ramsay (JRamsay(AT)wooster.edu), Jeffrey Willert (jawiller(AT)ncsu.edu), Hannah Roberts (HRoberts12(AT)wooster.edu), Nancy Tinoza (NTinoza12(AT)wooster.edu and Wenyuan Wu (wwu11(AT)wooster.edu), Sep 03 2009. The revisions are based on a dissection method found in 2009.
Given that the value of a(3) changed from 6 to 5 at the latest revision, one should not have too much confidence that these entries are minimal. - N. J. A. Sloane, Sep 05 2009
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STATUS
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approved
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