2,1

In an arbitrarily large pine plantation, a tree with a trunk of radius 1/n is located at each of the lattice points of a square lattice (whose rows are spaced one unit apart), except for one empty lattice point near the center of the plantation. For an observer located at the empty lattice point, how far away is the most distant visible tree trunk? The sequence a(n) is defined as the square of the distance from the observer to the most distant lattice point at which a visible tree trunk is located. (Each tree trunk is assumed to be a vertical cylinder, centered at its respective lattice point. A tree trunk is considered "visible" unless it is completely obscured from view by one or more other tree trunks.)

Jon E. Schoenfield, Table of n, a(n) for n = 2..3000

A different but related problem is addressed at Forests.

Example: at n = 5, there are 40 visible tree trunks; defining the origin as

the location of the observer, they are the ones located at (1,0), (4,1),

(3,1), (2,1), (3,2), (1,1), and all the additional locations that result from

using every possible reflection of them across the x-axis, the y-axis, or

the diagonal, y=x. (The tree trunk at (4,3) is considered completely obscured

by ones at (3,2) and (1,1), each of which is tangent to the line 4y = 3x.)

The most distant visible tree trunks are the ones located at the lattice point

(4,1) and its symmetrical locations; the square of their distance from the origin is 17, so a(5) = 17.

Cf. A124254, A124256.

Sequence in context: A031439 A074856 A087952 this_sequence A079936 A102854 A112634

Adjacent sequences: A124252 A124253 A124254 this_sequence A124256 A124257 A124258

nonn

Jon E. Schoenfield (jonscho(AT)hiwaay.net), Oct 22 2006