A193362 Numbers of ways in which a unit disc can be dissected into 6n curvilinear triangles, at least one of which does not contain the center

0, 31, 57, 99, 158, 237, 340, 472, 635, 836, 1075, 1361, 1696, 2087, 2538, 3054, 3641, 4306, 5053, 5891, 6822, 7857, 9000, 10260, 11643, 13156, 14807, 16605, 18556, 20671, 22954, 25418, 28069, 30918, 33973, 37243, 40738, 44469, 48444, 52676

Offset

1,2

References

H. T. Croft, K. J. Falconer and R. K. Guy, "Unsolved Problems in Geometry", 1991, page 89.

Links

Table of n, a(n) for n=1..40.

A. P. Goucher, Dissecting the disc, Complex Projective 4-Space.

Example

For n = 2, the a(2) = 31 dissections of the disc into 6n = 12 curvilinear triangles are:
* 1 solution in which 1 piece does not touch the center;
* 5 solutions in which 2 pieces do not touch the center;
* 10 solutions in which 3 pieces do not touch the center;
* 10 solutions in which 4 pieces do not touch the center;
* 3 solutions in which 5 pieces do not touch the center;
* 2 symmetrical solutions, one of which is exceptional.
The 30 non-exceptional cases are given in the article 'Dissecting the disc'.

Mathematica

Table[If[n==1, 0, Boole[n==2]+1+2 n+1+(3 n^2+3 n+2)/2+Floor[(2 n^3+6 n^2+7 n+6)/6]+Floor[(n^4+10 n^3+35 n^2+50 n+120)/120]+1], {n, 1, 100}]

Crossrefs

Sequence in context: A045116 A092227 A220539 * A293345 A096453 A218794

Adjacent sequences: A193359 A193360 A193361 * A193363 A193364 A193365

Keyword

nonn,easy

Author

Adam P. Goucher, Dec 20 2012

Status

approved