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

%I A141292 #10 Sep 30 2024 09:16:17
%S A141292 1,5,9,12,15,19,22,25,28,31,35,38,42,47,50,53,56,60,63,67,72,76,79,82,
%T A141292 85,89,93,100,103,106,109,113,117,121,126,130,133,136,139,143,147,156,
%U A141292 160,163,166,169,174,177,182,186,189,192,196,202,205,214,217,220,223
%N A141292 Conjectured values for minimal number of pieces required in a 2n-gon to square dissection that uses translation alone.
%D A141292 Boltyanskii, V.G., Equivalent and Equidecomposable Figures, D.C. Heath and Company, Boston, 1963.
%D A141292 G. N. Frederickson, Dissections Plane and Fancy, Camb. 1997.
%e A141292 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
%t A141292 b1 := (n, k) -> 2*sin(Pi*(2*k-1)/n)
%t A141292 b2 := (n, k) -> 2*sin(Pi*(2*k+1)/n)
%t A141292 w1 := (n, k) -> b1(n, k)+b2(n, k)
%t A141292 w2 := (n, k) -> sqrt((1/2)*n*sin(2*Pi/n))
%t A141292 h1 := (n, k) -> cos(Pi*(2*k-1)/n)-cos(Pi*(2*k+1)/n)
%t A141292 h2 := (n, k) -> w1(n, k)*h1(n, k)/w2(n, k)
%t A141292 a := (n, k) -> floor(w2(n, k)/w1(n, k))*h2(n, k)/h1(n, k)
%t A141292 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)))
%t A141292 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)
%t A141292 [seq([2*i, P(2*i)], i = 3 .. 100)]
%Y A141292 Cf. A110312.
%K A141292 nonn
%O A141292 2,2
%A A141292 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
%E A141292 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.
%E A141292 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