[seqfan] Re: a question about arrangements of pennies

Andrew Weimholt andrew.weimholt at gmail.com
Wed Dec 16 22:16:16 CET 2009

So far nobody has mentioned the idea of counting convex arrangements of pennies.

This appears to be in the OEIS as A116513, which is described as...

"Number of distinct hexagons of n points chosen from triangular
lattice A_2 with sides parallel to the principal axes of that lattice.
Degenerate sides (of length 1) are permitted."

It appears to me that the author intended these to be *convex* hexagons, with
degenerate sides of length *0* permitted. (A side of length 1 can
hardly be called degenerate).

A slightly simpler definition would be something like...
"Number of convex regions bound by edges of A_2 containing n vertices
of A_2 (in the interior or on the perimeter). Degenerate regions (with
area 0) are permitted."

The degenerate regions would be line segments of n vertices
(corresponding to n pennies in a row in the alternate definition).

Subtracting 1 from this sequence (i.e. removing the degenerate regions)
gives the Number of Convex Triangular Polyominoes containing n vertices of A_2.
(counting both vertices on the perimeter and in the interior of the polyominoe).
[not in OEIS].


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