A215010 Integer side lengths in arithmetic progression of simple convex hexagons with equal interior angles. Sequence gives the values of m for sides of lengths t+m*d, counterclockwise, for the two primitive solutions.

0, 5, 1, 3, 2, 4, 0, 5, 2, 1, 4, 3

Offset

1,2

Comments

Hexagons, such as (2,6,4,5,3,7), taken counterclockwise, can also be written (2,7,3,5,4,6) if considered clockwise or rotated 180 degrees and still read counterclockwise.

Conjecture: hexagons are the only simple convex polygons with equal interior angles with such property, due to the fact that cos(Pi/3) = 1/2.

The smaller such hexagon with all prime length sides is (7, 157, 67, 37, 127, 97). The smaller area of the two is sqrt(3)(6t^2 + 30td + 29d^2)/4 and the greater is sqrt(3)d^2/2 more.

Links

Table of n, a(n) for n=1..12.

Example

If first term is t = 1 and common difference is d = 1, we get (1, 6, 2, 4, 3, 5) and (1, 6, 3, 2, 5, 4); two hexagons with equal interior angles and all sides with consecutive integer lengths.
If t = 5 and d = 6 we get (5, 35, 11, 23, 17, 29) and (5, 35, 17, 11, 29, 23).

Crossrefs

Sequence in context: A055515 A363437 A338096 * A136744 A068237 A373363

Adjacent sequences: A215007 A215008 A215009 * A215011 A215012 A215013

Keyword

nonn,fini

Author

Robin Garcia, Jul 31 2012

Status

approved