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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).

%I A249870 #28 Feb 28 2025 23:12:33
%S A249870 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,
%T A249870 16,19,20,22,19,20,20,24,23,21,25,22,23,28,26,26,28,31,28,32,28,28,30,
%U A249870 32,34,35,32,33,38,34,36,38,37,40,37,38,43,40,44,40,46
%N 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).
%C A249870 The irrational parts are given in A249871.
%C A249870 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).
%C A249870 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.
%C A249870 For details see the notes given in a link.
%C A249870 This computation was inspired by a construction given by _Kival Ngaokrajang_ in A245094.
%H A249870 Wolfdieter Lang, <a href="/A249870/a249870_2.pdf">On lattice point circles for the Archimedean tiling (3, 4, 6, 4)</a>
%H A249870 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tiling_by_regular_polygons#Archimedean.2C_uniform_or_semiregular_tilings">Archimedean tilings</a>
%e A249870 The pairs [a(n), A249871(n)] for the squares of the radii R2(n) begin:
%e A249870 [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], ...
%e A249870 The corresponding radii R(n) are (Maple 10 digits, if not an integer):
%e A249870 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 ...
%Y A249870 Cf. A249871, A251627, A251628.
%K A249870 nonn,easy
%O A249870 0,3
%A A249870 _Wolfdieter Lang_, Dec 06 2014