2,2

a(2)=1 because a regular 4-gon-to-square dissection (using translations alone) can be accomplished with a single "piece". a(3) = 6 because a regular 6-gon-to-square dissection (using translations alone) can be accomplished with 6 pieces. Terms beyond a(3) are generated via an algorithm whose code is given below. [From Pamela Pierce (ppierce(AT)wooster.edu), Oct 16 2008]

a[n_, k_] := 2*(Cos[(2 k - 1)*Pi/n] + Cos[(2 k - 3)*Pi/n])

b[n_, k_] := Sqrt[n/2*Sin[2*Pi/n]]

c[n_, k_] := Sin[(2 k - 1)*Pi/n] - Sin[(2 k - 3)*Pi/n]

d[n_, k_] := a[n, k]*c[n, k]/b[n, k]

g[n_, k_] := 2*Cos[(2 k - 3)*Pi/n]

h[n_, k_] := 2*Cos[(2 k - 1)*Pi/n]

i[n_, k_] := -a[n, k] (l[n, k] - d[n, k])/d[n, k]

j[n_, k_] := l[n, k]/(-d[n, k]/a[n, k] + 2*c[n, k]/(-h[n, k] + g[n, k]))

l[n_, k_] := c[n, k] - Floor[b[n, k]/a[n, k]]*d[n, k]

m[n_, k_] := (l[n, k] - (2*c[n, k]*g[n, k])/(g[n, k] - h[n, k]))/(-2* c[n, k]/(g[n, k] - h[n, k]) - d[n, k]/a[n, k])

A[n_, k_] := (Sign[a[n, k] - 2*b[n, k]]/2 + 1/2)*(6 + Sign[b[n, k] - h[n, k]]/2 + 1/2)

B[n_, k_] := (Sign[a[n, k] - b[n, k]]/2 + 1/2)*(Sign[2*b[n, k] - a[n, k]]/2 + 1/2)*6

F[n_, k_] := (Sign[b[n, k] - a[n, k]]/2 + 1/2)*(3*(Floor[b[n, k]/a[n, k]] - 1) + 7 + (Sign[m[n, k] - i[n, k]])/2 + 1 + Sign[-i[n, k] + j[n, k]]/2)

p[n_, k_] := A[n, k] + B[n, k] + F[n, k]

P[n_] := 3 + Sum_(k = 2)^(Ceiling[2*n/4]) (p[2*n, k])

Cf. A110312.

Sequence in context: A007262 A132107 A129317 this_sequence A020183 A039280 A045091

Adjacent sequences: A141289 A141290 A141291 this_sequence A141293 A141294 A141295

nonn

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

2nd term was edited because we found a hexagon-to-square dissection using translations alone which uses only 6 pieces. Pamela Pierce (ppierce(AT)wooster.edu), Oct 16 2008