A243924 Irregular triangular array of taxicab norms of Gaussian integers in array G generated as at Comments.

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, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 6, 6, 6

Offset: 1

Author

Clark Kimberling, Jun 17 2014

Keywords

• nonn
• easy
• tabf

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.

Examples

			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
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A233694, A226080.

Programs

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 *)