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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 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

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