A256060 Queen Dido's puzzle (the founding of Carthage): a(n) is twice the maximal area of a polygon with 1) vertices on integral Cartesian coordinates, 2) no two edges parallel, and 3) all edge lengths less than or equal to n^2.

0, 0, 1, 1, 2, 36, 36, 36, 50, 53, 153, 153, 153, 333, 333, 333, 360

Offset

0,5

Comments

The sequence may increase when n is the sum of two squares (A001481).

An optimal polygon will always be convex. - Gordon Hamilton

For parity reasons, the edges of the maximal-area polygon are not always as long as possible. This is true for a(9) through a(12). - Gordon Hamilton

This puzzle sequence could be used when introducing students to slopes.

Are these values known to be optimal or are they conjectures? - N. J. A. Sloane, Mar 13 2015

These values have not been proved to be optimal.

Links

Table of n, a(n) for n=0..16.

Example

a(4) = 2 because this triangle has area 1 (remember a(n) is twice the area):
                          . . . . .
                          . x . x .
                          . . x . .
                          . . . . .
a(5) = a(6) = a(7) = 36 because of this polygon of area 18:
                      . . . . . . . .
                      . . x . x . . .
                      . . . . . x . .
                      . x . . . . . .
                      . . . . . . x .
                      . x . . . x . .
                      . . . x . . . .
                      . . . . . . . .
a(8) = 50 because of this polygon of area 25:
                     . . . . . . . . .
                     . . . . . . . . .
                     . . . x . x . . .
                     . x . . . . . . .
                     . . . . . . . x .
                     . x . . . . . . .
                     . . . . . . x . .
                     . . x . . . . . .
                     . . . . x . . . .
                     . . . . . . . . .
a(9) = 53 because of this polygon of area 26.5:
                     . . . . . . . . .
                     . . . x . . . . .
                     . x . . . x . . .
                     . . . . . . . . .
                     . . . . . . . x .
                     . x . . . . . . .
                     . . . . . . x . .
                     . . x . . x . . .
                     . . . . . . . . .
a(10) = 153 because of this polygon of area 76.5:
                  . . . . . . . . . . . . .
                  . . . x . . x . . . . . .
                  . . x . . . . . x . . . .
                  . . . . . . . . . . . . .
                  . x . . . . . . . . x . .
                  . . . . . . . . . . . . .
                  . . . . . . . . . . . x .
                  . x . . . . . . . . . . .
                  . . . . . . . . . . . . .
                  . . . . . . . . . . x . .
                  . . x . . . . . x . . . .
                  . . . . . x . . . . . . .
                  . . . . . . . . . . . . .

Crossrefs

Sequence in context: A095397 A353216 A073406 * A096513 A037418 A239343

Adjacent sequences: A256057 A256058 A256059 * A256061 A256062 A256063

Keyword

nonn,more

Author

Gordon Hamilton, Mar 13 2015

Status

approved