A251629 Rational parts of the Q(sqrt(2)) integers giving the squared radii of the lattice point circles for the Archimedean tiling (4,8,8).

0, 1, 2, 2, 3, 4, 5, 6, 6, 7, 9, 9, 11, 10, 12, 12, 13, 14, 14, 15, 17, 18, 17, 18, 21, 22, 20, 22, 22, 25, 23, 24, 25, 27, 28, 29, 29, 30, 30, 33, 34, 34, 33, 35, 36, 34, 39, 38, 37, 41, 39, 42, 41, 44, 42, 43, 44, 46, 46, 49, 48, 50, 49

Offset: 0

Author

Wolfdieter Lang, Jan 02 2015

Keywords

• nonn
• easy

Comments

The irrational parts are given in A251631.

The points of the lattice of the Archimedean tiling (4,8,8) lie on certain circles around any point. The length of the regular octagon (8-gon) side 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(2)), hence R2(n) = a(n) + A251631(n)*sqrt(2). The R2 sequence is sorted in increasing order.

For the case of the Archimedean tiling (3,4,6,4) see A249870 and A249871, and the W. Lang link given in A249870.

Examples

			The first pairs [a(n), A251631(n)] for the squared radii are: [0,0], [1,0], [2,0], [2,1], [3,2], [4,2], [5,2] [6,3], [6,4], [7,4], [9,4], [9,6], [11,6], [10,7], [12,6], [12,8], [13,8], ...
The corresponding radii are (Maple 10 digits if not integer) 0, 1, 1.414213562, 1.847759065, 2.414213562, 2.613125930, 2.797932652, 3.200412581, 3.414213562, 3.557647291, 3.828427124, 4.181540551, 4.414213562, 4.460884994, 4.526066876, 4.828427124, 4.930893276, ...
		

Links

• Wolfdieter Lang, On lattice point circles for the Archimedean tiling (4,8,8).
• Wikipedia, Archimedean tilings

Crossrefs

Cf. A251631, A251632, A251633.

Cf. A249870, A249871 ((3,4,6,4) tiling).