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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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified June 24 13:17 EDT 2019. Contains 324325 sequences. (Running on oeis4.)