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A322041 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. 2
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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

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.

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).

REFERENCES

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)).

LINKS

Table of n, a(n) for n=1..91.

N. J. A. Sloane, Rows 1 through 36.

N. J. A. Sloane, Illustration for n=2

N. J. A. Sloane, Illustration for n=3

N. J. A. Sloane, Illustration for n=4

N. J. A. Sloane, Illustration for n=6

N. J. A. Sloane, Illustration for n=8

FORMULA

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.

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

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)].

For t=3, for example, we get:

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

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

EXAMPLE

The first 18 rows are

1 [1]

2 [1, 3]

3 [1, 6, 2]

4 [1, 6, 9, 0]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

MAPLE

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

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

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

for i from 0 to n-1 do

for j from 0 to n-1 do

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

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

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

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

fi;

   A[m]:=A[m]+1;

od: od:

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

end;

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

CROSSREFS

The rows converge to A008458.

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

Sequence in context: A067231 A218971 A145791 * A130302 A270103 A134546

Adjacent sequences:  A322038 A322039 A322040 * A322042 A322043 A322044

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Dec 05 2018; corrected and extended Dec 06 2018

STATUS

approved

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Last modified September 16 21:28 EDT 2021. Contains 347473 sequences. (Running on oeis4.)