

A124255


Forestandtrees problem: square of distance to most distant visible tree.


3



2, 5, 13, 17, 34, 41, 61, 74, 97, 113, 137, 157, 194, 221, 250, 281, 317, 353, 397, 433, 482, 521, 569, 617, 674, 725, 778, 829, 898, 953, 1021, 1082, 1154, 1217, 1289, 1361, 1433, 1517, 1597, 1669, 1762, 1825, 1933, 2018, 2113, 2197, 2297, 2393, 2498, 2594
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OFFSET

2,1


COMMENTS

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.)


LINKS

Jon E. Schoenfield, Table of n, a(n) for n = 2..3000
A different but related problem is addressed at Forests.


EXAMPLE

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 xaxis, the yaxis, 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.


CROSSREFS

Cf. A124254, A124256.
Sequence in context: A031439 A074856 A087952 * A293173 A079936 A102854
Adjacent sequences: A124252 A124253 A124254 * A124256 A124257 A124258


KEYWORD

nonn


AUTHOR

Jon E. Schoenfield, Oct 22 2006


STATUS

approved



