This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For both n = 0 and n = 1, the a(0) = a(1) = 2 points are the initial points: (0,0) and (1,0). For n = 2, the a(2) = 6 points are the two above points along with (-1,0), (2,0), (1/2,-sqrt(3)/2), and (1/2,sqrt(3)/2): (-1,0): formed by intersecting the line between (0,0) and (1,0) with the circle of radius 1 centered at (0,0). (2,0): formed by intersecting the line between (0,0) and (1,0) with the circle of radius 1 centered at (1,0). (1/2,-sqrt(3)/2), and (1/2,sqrt(3)/2): formed by intersecting the circle of radius 1 centered at (0,0) with the circle of radius 1 centered at (1,0).
For n = 0 and n = 1, the only number that is constructible is 1, the distance between the two initial points. For n = 2, we additionally can construct sqrt(3) and 2. To construct sqrt(3), draw two unit circles, centered at each of the two starting points. These unit circles intersect in two places, which are a distance of sqrt(3) apart. To construct 2, draw a unit circle along with the line connecting the starting points. The line marks two points that are opposite of each other on the unit circle. For n = 3, we additionally can construct 3 and 4.
n\k | 0 1 2 3 4 5 ----+-------------------------- 0 | 1 1 | 1 2 2 | 0 2 1 3 | 0 0 12 4 4 | 0 0 45 116 44 5 | 0 0 232 1565 3005 1084 6 | 0 0 1627 34114 166556 249494 91192 T(2,1) = 2 because there are two ruler-and-compass constructions with a line and a circle: A circle centered at (0,0) through (1,0) and a line through (0,0),(1,0). A circle centered at (1,0) through (0,0) and a line through (0,0),(1,0). T(2,2) = 1 because there is one ruler-and-compass construction with two circles: A circle centered at (0,0) through (1,0) and a circle centered at (1,0) through (0,0).
For example the following two constructions are considered the same: (1) Draw a circle centered at (0,0) through (1,0), and then draw a line through (0,0) and (1,0). (2) Draw a line through (0,0) and (0,1) and then draw a circle centered at (1,0) through (0,0).
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