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A305185 a(n) minimizes the maximum norm of elements in a complete residue system of Eisenstein integers modulo n. 1
0, 1, 3, 4, 7, 12, 13, 19, 27, 28, 37, 48, 49, 61, 75, 76, 91, 108, 109, 127, 147, 148, 169, 192, 193, 217, 243, 244, 271, 300, 301, 331, 363, 364, 397, 432, 433, 469, 507, 508, 547, 588, 589, 631, 675, 676, 721, 768, 769, 817, 867, 868, 919, 972, 973, 1027, 1083, 1084, 1141, 1200 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

From Jianing Song, May 05 2019:

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.

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.

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)

LINKS

Jianing Song, Table of n, a(n) for n = 1..10000

Jianing Song, Further notes on A305185

FORMULA

From Jianing Song, May 05 2019: (Start)

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.

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.

G.f.: (x^2*(1 + x^2)*(1 + 2*x - x^3 + x^4))/((1 - x)^3*(1 + x + x^2)^2) (End)

EXAMPLE

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}.

n = 1: S(1) = {0}, so a(1) = max_{s in S(1)} N(s) = 0.

n = 2: S(2) = {0, 1, w, w+1}, so a(2) = max_{s in S(2)} N(s) = 1.

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.

PROG

(PARI) a(n) = if(n>1, n^2 - 3*n*ceil(n/3) + 3*ceil(n/3)^2, 0) \\ Jianing Song, May 12 2019

CROSSREFS

Cf. A004016, A033687.

Sequence in context: A286728 A300332 A244819 * A083561 A018195 A086417

Adjacent sequences:  A305182 A305183 A305184 * A305186 A305187 A305188

KEYWORD

nonn,easy

AUTHOR

Jianing Song, May 27 2018

EXTENSIONS

Entry rewritten by Jianing Song, May 05 2019

STATUS

approved

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Last modified September 19 00:22 EDT 2021. Contains 347549 sequences. (Running on oeis4.)