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A141292 Conjectured values for minimal number of pieces required in a 2n-gon to square dissection that uses translation alone. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

REFERENCES

Boltyanskii, V.G., Equivalent and Equidecomposable Figures, D.C. Heath and Company, Boston, 1963.

G. N. Frederickson, Dissections Plane and Fancy, Camb. 1997.

LINKS

Table of n, a(n) for n=2..60.

EXAMPLE

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

MATHEMATICA

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)]

CROSSREFS

Cf. A110312.

Sequence in context: A153044 A106635 A182617 * A186512 A068477 A190702

Adjacent sequences:  A141289 A141290 A141291 * A141293 A141294 A141295

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified August 20 04:36 EDT 2014. Contains 245796 sequences.