

A243924


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


4



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
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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 4n13 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 . 1i . 1i
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

Cf. A233694, A226080.
Sequence in context: A105102 A105105 A178677 * A272759 A272760 A054717
Adjacent sequences: A243921 A243922 A243923 * A243925 A243926 A243927


KEYWORD

nonn,easy,tabf


AUTHOR

Clark Kimberling, Jun 17 2014


STATUS

approved



