A383086 The number of distinct distances between points in the Euclidean plane where the points are constructed via a straightedge-and-compass construction using n circles and no lines.

1, 1, 2, 4, 35, 2480

Offset

0,3

Comments

We say that a real number is a constructible number if it is the distance between two points that can be determined from a straightedge-and-compass construction.

A straightedge-and-compass construction starts with 2 points marked on the plane, traditionally (0,0) and (1,0). In the constructions counted by this sequence, only the compass is used. Circles can be drawn at any marked point through any other marked point, and new points are marked where circles intersect.

Links

Table of n, a(n) for n=0..5.

Wikipedia, Constructible number

Wikipedia, Mohr-Mascheroni theorem

Example

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): 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.
For n = 3, we additionally can construct 2, and 3.

Crossrefs

Cf. A383083, A383085, A383087.

Sequence in context: A348094 A140984 A057996 * A009090 A009295 A396049

Adjacent sequences: A383083 A383084 A383085 * A383087 A383088 A383089

Keyword

nonn,more,hard

Author

Peter Kagey, Apr 16 2025

Status

approved