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%I #58 Jul 29 2020 05:38:23
%S 0,1,3,4,7,12,13,19,27,28,37,48,49,61,75,76,91,108,109,127,147,148,
%T 169,192,193,217,243,244,271,300,301,331,363,364,397,432,433,469,507,
%U 508,547,588,589,631,675,676,721,768,769,817,867,868,919,972,973,1027,1083,1084,1141,1200
%N a(n) minimizes the maximum norm of elements in a complete residue system of Eisenstein integers modulo n.
%C From _Jianing Song_, May 05 2019:
%C For any Eisenstein integer w != 0, let R(w) be any set of N(w) Eisenstein integers such that no two numbers are congruent modulo w, then we intend to find the smallest possible value of max_{s in R(w)} N(s). Here N(w) is the norm of w.
%C If we can find a set of complex numbers A such that: (i) for any Eisenstein integer x, r in A, |r| <= |r - x|; (ii) every complex number z can be uniquely represented as z = x + r, where x is an Eisenstein integer, r is in A, then S(w) = {r*w : r is in A} is a complete residue system modulo w formed by choosing one element with the minimal norm in each residue class modulo w (there may be more than one element whose norms are minimal in one residue class). As a result, the smallest possible value of max_{s in R(w)} N(s) is max_{s in S(w)} N(s). For more details, see my further notes in the Link section.
%C Now, for positive integers n, we find the value of max_{s in S(n)} N(s) over the ring of Eisenstein integers. Let A be the set shown in Page 5, Figure 2 in my further notes on this sequence (see Links section below), and S(w) = {r*w : r in A}. For n >= 2, note that for any s in S(n), s != 0, there exists some s' in S(n) such that s/s' is an Eisenstein unit and arg(s') is in the range [-Pi/6, Pi/6]. Let s' = (x + y*sqrt(3)*i)/2 where x and y have the same parity, 0 < x <= n and -x/3 <= y <= x/3, then N(s) = N(s') = (x^2 + 3*y^2)/4. For fixed x >= 2, we have max |y| = x - 2*ceiling(x/3) so max N(s') = max_{x=2..n} (x^2 + 3*(x - 2*ceiling(x/3))^2)/4 = (n^2 + 3*(n - 2*ceiling(n/3))^2)/4. (End)
%H Jianing Song, <a href="/A305185/b305185.txt">Table of n, a(n) for n = 1..10000</a>
%H Jianing Song, <a href="/A305185/a305185_2.pdf">Further notes on A305185</a>
%F From _Jianing Song_, May 05 2019: (Start)
%F a(1) = 0; for n >= 2, a(n) = (n^2 + 3*(n - 2*ceiling(n/3))^2)/4 = n^2 - 3*n*ceiling(n/3) + 3*ceiling(n/3)^2.
%F For k >= 1, a(3*k-1) = 3*k^2 - 3*k + 1, a(3*k) = 3*k^2, a(3*k+1) = 3*k^2 + 1.
%F G.f.: (x^2*(1 + x^2)*(1 + 2*x - x^3 + x^4))/((1 - x)^3*(1 + x + x^2)^2) (End)
%e In the following examples let w = (-1 + sqrt(-3))/2. Let A be the set shown in Page 5, Figure 2 in my further notes on this sequence, and S(w) = {r*w : r is in A}.
%e n = 1: S(1) = {0}, so a(1) = max_{s in S(1)} N(s) = 0.
%e n = 2: S(2) = {0, 1, w, w+1}, so a(2) = max_{s in S(2)} N(s) = 1.
%e n = 3: S(3) = {0, 1, -1, w, w+1, -w, -w-1, w+2, -w-2}, so a(3) = max_{s in S(3)} N(s) = 3.
%o (PARI) a(n) = if(n>1, n^2 - 3*n*ceil(n/3) + 3*ceil(n/3)^2, 0) \\ _Jianing Song_, May 12 2019
%Y Cf. A004016, A033687.
%K nonn,easy
%O 1,3
%A _Jianing Song_, May 27 2018
%E Entry rewritten by _Jianing Song_, May 05 2019
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