A249870 Rational parts of the Q(sqrt(3)) integers giving the square of the radii for lattice point circles for the Archimedean tiling (3, 4, 6, 4).

0, 1, 2, 3, 2, 4, 4, 4, 5, 6, 8, 8, 7, 8, 10, 10, 10, 13, 14, 11, 12, 13, 15, 14, 16, 16, 17, 16, 19, 20, 22, 19, 20, 20, 24, 23, 21, 25, 22, 23, 28, 26, 26, 28, 31, 28, 32, 28, 28, 30, 32, 34, 35, 32, 33, 38, 34, 36, 38, 37, 40, 37, 38, 43, 40, 44, 40, 46

Offset

0,3

Comments

The irrational parts are given in A249871.

The points of the lattice of the Archimedean tiling (3, 4, 6, 4) lie on certain circles around any point. The length of the side of the regular 6-gon is taken as 1 (in some length unit).

The squares of the radii R2(n) of these circles are integers in the real quadratic number field Q(sqrt(3)), hence R2(n) = a(n) + A249871(n)*sqrt(3). The R2 sequence is sorted in increasing order.

For details see the notes given in a link.

This computation was inspired by a construction given by Kival Ngaokrajang in A245094.

Links

Table of n, a(n) for n=0..67.

Wolfdieter Lang, On lattice point circles for the Archimedean tiling (3, 4, 6, 4)

Wikipedia, Archimedean tilings

Example

The pairs [a(n), A249871(n)] for the squares of the radii R2(n) begin:
[0, 0], [1, 0], [2, 0], [3, 0], [2, 1], [4, 0], [4, 1], [4, 2], [5, 2], [6, 3], [8, 2], [8, 3], [7, 4], [8, 4], [10, 3]] ...
The corresponding radii R(n) are (Maple 10 digits, if not an integer):
0, 1, 1.414213562, 1.732050808, 1.931851653, 2, 2.394170171, 2.732050808, 2.909312911, 3.346065215, 3.385867927, 3.632650881, 3.732050808, 3.863703305, 3.898224265 ...

Crossrefs

Cf. A249871, A251627, A251628.

Sequence in context: A322348 A081315 A035662 * A325404 A324750 A320348

Adjacent sequences: A249867 A249868 A249869 * A249871 A249872 A249873

Keyword

nonn,easy

Author

Wolfdieter Lang, Dec 06 2014

Status

approved