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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
0, 5, 1, 3, 2, 4, 0, 5, 2, 1, 4, 3 (list; graph; refs; listen; history; text; internal format)
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: A206076 A115638 A055515 * A136744 A068237 A083345

Adjacent sequences:  A215007 A215008 A215009 * A215011 A215012 A215013

KEYWORD

nonn,fini

AUTHOR

Robin Garcia, Jul 31 2012

STATUS

approved

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Last modified February 16 18:53 EST 2019. Contains 320165 sequences. (Running on oeis4.)