A141292 Conjectured values for minimal number of pieces required in a 2n-gon to square dissection that uses translation alone.

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

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

Keywords

• nonn

Examples

			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
		

References

• Boltyanskii, V.G., Equivalent and Equidecomposable Figures, D.C. Heath and Company, Boston, 1963.
• G. N. Frederickson, Dissections Plane and Fancy, Camb. 1997.

Crossrefs

Cf. A110312.

Programs

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)]

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