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A243924 Irregular triangular array of taxicab norms of Gaussian integers in array G generated as at Comments. 4

%I #11 Jun 19 2014 11:17:55

%S 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,

%T 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,

%U 5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,6,6,6

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

%C 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.

%H Clark Kimberling, <a href="/A243924/b243924.txt">Table of n, a(n) for n = 1..2000</a>

%e First 6 rows of G:

%e 0

%e 1

%e 2 .. i

%e 3 .. 2i .. i+1 ... -1

%e 4 .. 3i .. 1+2i .. -2 .. i+2 .. -1+i . -i

%e 5 .. 4i .. 1+3i .. -3 .. 2+2i . -2+i . -2i . i+3 . -1+2i . -1-i . 1-i

%e The corresponding taxicab norms follow:

%e 0

%e 1

%e 1 2

%e 1 2 2 3

%e 2 2 1 3 3 3 4

%e 3 3 2 3 2 4 2 4 4 4 5

%e Each row is then arranged in nondecreasing order:

%e 0

%e 1

%e 1 2

%e 1 2 2 3

%e 1 2 2 3 3 3 4

%e 2 2 2 3 3 3 4 4 4 4 5

%t z = 10; g[1] = {0}; f1[x_] := x + 1; f2[x_] := I*x; h[1] = g[1];

%t b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];

%t h[n_] := h[n] = Union[h[n - 1], g[n - 1]];

%t g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]

%t Table[g[n], {n, 1, z}] (* the array G *)

%t v = Table[Abs[Re[g[n]]] + Abs[Im[g[n]]], {n, 1, z}]

%t w = Map[Sort, v] (* A243924, rows *)

%t w1 = Flatten[w] (* A243924, sequence *)

%Y Cf. A233694, A226080.

%K nonn,easy,tabf

%O 1,4

%A _Clark Kimberling_, Jun 17 2014

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