A322041 - OEIS

%I #56 Sep 26 2019 02:32:07

%S 1,1,3,1,6,2,1,6,9,0,1,6,12,6,0,1,6,12,15,2,0,1,6,12,18,12,0,0,1,6,12,

%T 18,21,6,0,0,1,6,12,18,24,18,2,0,0,1,6,12,18,24,27,12,0,0,0,1,6,12,18,

%U 24,30,24,6,0,0,0,1,6,12,18,24,30,33,18,2,0,0,0,1,6,12,18,24,30,36,30,12,0,0,0,0

%N Triangle read by rows: let E denote the standard triangular 6-valent grid in the plane, regarded as a graph with the Eisenstein integers as vertices; row n gives the coordination sequence of the quotient graph E/nE.

%C The Eisenstein integers E are the complex numbers r+s*omega, where r, s in Z and omega = exp(2*Pi*i/3) is a complex cube root of unity.

%C Denote the entries in the triangle by T(n,k), for n >= 1, 0 <= k <= n-1. Then T(n,k) <= 6*k for k >= 1, and Sum_{k=0..n-1} T(n,k) = n^2.

%C When E is regarded as a lattice in R^2, E/nE has packing radius roughly n/2, but covering radius roughly n/sqrt(3) > n/2 (see Conway-Sloane, Chapter 4). This means that as n increases, the number of terms in the n-th row of the triangle will increase linearly with n. The largest k such that T(n,k) is nonzero is A322042(n), which is conjecturally n - ceiling(n/3).

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993. Fig. 7.1, p. 199. Illustrates row 2 (note that E/2E is isomorphic to GF(4)).

%H N. J. A. Sloane, <a href="/A322041/a322041_1.txt">Rows 1 through 36.</a>

%H N. J. A. Sloane, <a href="/A322041/a322041.png">Illustration for n=2</a>

%H N. J. A. Sloane, <a href="/A322041/a322041_1.png">Illustration for n=3</a>

%H N. J. A. Sloane, <a href="/A322041/a322041_2.png">Illustration for n=4</a>

%H N. J. A. Sloane, <a href="/A322041/a322041_4.png">Illustration for n=6</a>

%H N. J. A. Sloane, <a href="/A322041/a322041_3.png">Illustration for n=8</a>

%F Examination of the first 36 rows (see link) shows an obvious quasi-periodic structure. Call an entry T(n,k) "full" if k=0 or T(n,k)=6*k. Then it appears that column k>0 is full starting at n=2k+1. It also appears that the number of trailing 0's is floor((n-1)/3) (see A322042). Combining these two observations suggests that the rows of the triangle are quasi-periodic with period 6.

%F One can now formulate a specific conjecture for what row n is, for each of the six residue classes of n mod 6.

%F For example, suppose n=6t. Then it appears that row n is [1, 6, 18, 24, ..., 18t-6, 18t-3, 18(t-1), 18(t-2), 18(t-3), ..., 36, 18, 2, 0 (2t-1 times)].

%F For t=3, for example, we get:

%F [1, 6, 12, 18, 24, 30, 36, 42, 48, 51, 36, 18, 2, 0, 0, 0, 0, 0]

%F There are similar conjectures for n = 6t+1, ..., 6t+5.

%e The first 18 rows are

%e 1 [1]

%e 2 [1, 3]

%e 3 [1, 6, 2]

%e 4 [1, 6, 9, 0]

%e 5 [1, 6, 12, 6, 0]

%e 6 [1, 6, 12, 15, 2, 0]

%e 7 [1, 6, 12, 18, 12, 0, 0]

%e 8 [1, 6, 12, 18, 21, 6, 0, 0]

%e 9 [1, 6, 12, 18, 24, 18, 2, 0, 0]

%e 10 [1, 6, 12, 18, 24, 27, 12, 0, 0, 0]

%e 11 [1, 6, 12, 18, 24, 30, 24, 6, 0, 0, 0]

%e 12 [1, 6, 12, 18, 24, 30, 33, 18, 2, 0, 0, 0]

%e 13 [1, 6, 12, 18, 24, 30, 36, 30, 12, 0, 0, 0, 0]

%e 14 [1, 6, 12, 18, 24, 30, 36, 39, 24, 6, 0, 0, 0, 0]

%e 15 [1, 6, 12, 18, 24, 30, 36, 42, 36, 18, 2, 0, 0, 0, 0]

%e 16 [1, 6, 12, 18, 24, 30, 36, 42, 45, 30, 12, 0, 0, 0, 0, 0]

%e 17 [1, 6, 12, 18, 24, 30, 36, 42, 48, 42, 24, 6, 0, 0, 0, 0, 0]

%e 18 [1, 6, 12, 18, 24, 30, 36, 42, 48, 51, 36, 18, 2, 0, 0, 0, 0, 0]

%e ...

%p # We work in a fundamental region for E/nE and calculate the edge-distance of each point to the nearest point of nE.

%p hist:=proc(n) local A,i,j,m,d1,d2,d3,d4;

%p A:=Array(0..n,0);

%p for i from 0 to n-1 do

%p for j from 0 to n-1 do

%p d1:=i+j; d2:=n-i; d3:=2*n-i-j; d4:=n-j;

%p if i+j<n then m:=min(d1,d2,d3,d4);

%p elif i+j=n then m:=min(i,j);

%p else m:=min(i,j,d1,d3);

%p fi;

%p    A[m]:=A[m]+1;

%p od: od:

%p [seq(A[i],i=0..n-1)];

%p end;

%p for n from 1 to 14 do lprint(hist(n)); od:

%Y The rows converge to A008458.

%Y Cf. A322038 (an analog for the square grid), A322042.

%K nonn,tabl

%O 1,3

%A _N. J. A. Sloane_, Dec 05 2018; corrected and extended Dec 06 2018