OFFSET
1,4
COMMENTS
An array G of Gaussian integers is generated as follows: (row 1) = (0), and for n >=2, row n consists of the numbers x+1 and then i*x, where duplicates are deleted as they occur. Every Gaussian integer occurs exactly once in G. The taxicab norm of a Gaussian integer b+c*i is the taxicab distance (also known as Manhattan distance) from 0 to b+c*i, given by |b|+|c|. The norms of numbers in row n are given here in nondecreasing order. Conjecture: the number of numbers in row n is 4n-13 for n >= 5.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..2000
EXAMPLE
First 6 rows of G:
0
1
2 .. i
3 .. 2i .. i+1 ... -1
4 .. 3i .. 1+2i .. -2 .. i+2 .. -1+i . -i
5 .. 4i .. 1+3i .. -3 .. 2+2i . -2+i . -2i . i+3 . -1+2i . -1-i . 1-i
The corresponding taxicab norms follow:
0
1
1 2
1 2 2 3
2 2 1 3 3 3 4
3 3 2 3 2 4 2 4 4 4 5
Each row is then arranged in nondecreasing order:
0
1
1 2
1 2 2 3
1 2 2 3 3 3 4
2 2 2 3 3 3 4 4 4 4 5
MATHEMATICA
z = 10; g[1] = {0}; f1[x_] := x + 1; f2[x_] := I*x; h[1] = g[1];
b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
Table[g[n], {n, 1, z}] (* the array G *)
v = Table[Abs[Re[g[n]]] + Abs[Im[g[n]]], {n, 1, z}]
w = Map[Sort, v] (* A243924, rows *)
w1 = Flatten[w] (* A243924, sequence *)
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Clark Kimberling, Jun 17 2014
STATUS
approved