|
Search: id:A141292
|
|
|
| A141292 |
|
Conjectured values for minimal number of pieces required in a 2n-gon to square dissection that uses translation alone. |
|
+0 1
|
|
| 1, 6, 9, 17, 15, 26, 23, 33, 29, 41, 35, 48, 44, 58, 51, 64, 57, 74, 64, 83, 74, 91, 80, 100, 86, 110, 96, 119, 103, 125, 111, 137, 118, 144, 127, 155, 134, 164, 141, 172, 151, 182, 160, 190, 168, 202, 175, 208, 183, 218, 192, 229, 199, 239, 206
(list; graph; listen)
|
|
|
OFFSET
|
2,2
|
|
|
EXAMPLE
|
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]
|
|
MATHEMATICA
|
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])
|
|
CROSSREFS
|
Cf. A110312.
Sequence in context: A007262 A132107 A129317 this_sequence A020183 A039280 A045091
Adjacent sequences: A141289 A141290 A141291 this_sequence 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
|
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
|
|
|
Search completed in 0.002 seconds
|