A124254 Forest-and-trees problem: number of trees visible. (See Comments.)

1, 2, 4, 5, 9, 11, 15, 19, 24, 28, 33, 38, 48, 55, 60, 67, 77, 84, 96, 104, 116, 125, 139, 148, 160, 173, 186, 197, 213, 227, 245, 259, 278, 293, 310, 324, 344, 364, 383, 397, 420, 435, 462, 482, 502, 522, 549, 572, 597, 622, 648, 669, 696, 720, 750, 774, 802

Offset: 2

Author

Jon E. Schoenfield, Oct 22 2006

Keywords

• nonn

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 at the center of the plantation. How many tree trunks will be visible to an observer located at the empty lattice point? By symmetry, the number must always be divisible by 8, so a(n) is defined to be one-eighth of the number of visible tree trunks. (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.)

Conjecture: a(n) is the number of pairs of coprime numbers (x,y) such that x <= y and x^2 + y^2 < n^2. The Mathematica program is based on this assumption. How important is the decreasing function 1/n of the trunk's radius with respect to n? Does the sequence change if this function changes? - Andres Cicuttin, Feb 24 2023

Examples

			Example: at n = 5, there are 8*a(5) = 40 visible tree trunks; defining the origin as the location of the observer, 4 are centered at points on the axes (i.e., (1,0), (0,1), (-1,0) and (0,-1)), 4 are at points on the diagonals (i.e., (1,1), (-1,1), (-1,-1) and (1,-1)) and the remaining 32, beginning in counterclockwise order from the +x-axis, are the ones at (4,1), (3,1), (2,1), (3,2) and the 28 others that result from using every possible reflection of those points across the x-axis, the y-axis, or the diagonal, y=x. (The tree trunk at (4,3) is considered completely obscured by the ones at (3,2) and (1,1), each of which is tangent to the line 4y = 3x.)
		

Links

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

Crossrefs

Cf. A124255, A124256.

Programs

Mathematica

pairs[n_] := Flatten[Table[Table[{i, j}, {i, 1, j}], {j, 1, n}], 1];
Table[(Length@Select[pairs[j], And[GCD[#[[1]], #[[2]]] == 1 , #[[1]]^2 + #[[2]]^2 < j^2] &]), {j, 2, 70}] (* Andres Cicuttin, Feb 24 2023 *)

Formula

Does a(n)/n^2 approach 0.75/Pi?