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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
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.
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: A182617 A314615 A314616 * A186512 A314617 A068477
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