The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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: A182617 A314615 A314616 * A186512 A314617 A068477 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 10 06:17 EDT 2024. Contains 375773 sequences. (Running on oeis4.)