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A366353 a(0) = 0; for n > 0, a(n) is the largest taxicab distance on a square spiral between a(n-1) and any previous occurrence of a(n-1). If a(n-1) has not previously occurred then a(n) = 0.

%I A366353 #12 Oct 16 2023 13:43:02
%S A366353 0,0,1,0,2,0,2,2,3,0,4,0,4,2,5,0,6,0,6,2,6,4,7,0,6,8,0,7,5,4,8,5,3,4,
%T A366353 6,8,10,0,9,0,7,7,8,12,0,7,6,8,10,12,6,10,11,0,9,8,13,0,11,6,9,6,11,
%U A366353 10,13,8,12,13,11,10,9,12,8,15,0,13,13,12,11,12,13,16,0,13,15,11,11,10,12
%N A366353 a(0) = 0; for n > 0, a(n) is the largest taxicab distance on a square spiral between a(n-1) and any previous occurrence of a(n-1). If a(n-1) has not previously occurred then a(n) = 0.
%H A366353 Scott R. Shannon, <a href="/A366353/b366353.txt">Table of n, a(n) for n = 0..10000</a>
%H A366353 Scott R. Shannon, <a href="/A366353/a366353.png">Image of the first 500000 terms</a>.
%H A366353 Scott R. Shannon, <a href="/A366353/a366353_1.png">Image of the first 50000 terms on the square spiral</a>. The colors are graduated across the spectrum to show their relative size. Zoom in to see the numbers.
%e A366353 The spiral begins:
%e A366353 .
%e A366353                                 .
%e A366353     10--8---6---4---3---5---8   :
%e A366353     |                       |   :
%e A366353     0   6---0---5---2---4   4   9
%e A366353     |   |               |   |   |
%e A366353     9   0   2---0---1   0   5   0
%e A366353     |   |   |       |   |   |   |
%e A366353     0   6   0   0---0   4   7   11
%e A366353     |   |   |           |   |   |
%e A366353     7   2   2---2---3---0   0   10
%e A366353     |   |                   |   |
%e A366353     7   6---4---7---0---6---8   6
%e A366353     |                           |
%e A366353     8---12--0---7---6---8---10--12
%e A366353 .
%e A366353 a(2) = 1 as the taxicab distance between a(1) = 0, at (1,0) relative to the starting square, and the only previous occurrence of 0, a(0) at (0,0), is 1.
%e A366353 a(8) = 3 as the maximum taxicab distance between a(7) = 2, at (0,-1) relative to the starting square, and any previous occurrence of 2 is 3, to a(4) = 2 at (-1,1) relative to the starting square.
%e A366353 a(32) = 3 as the maximum taxicab distance between a(31) = 5, at (2,3) relative to the starting square, and any previous occurrence of 5 is 3, to a(28) = 5 at (3,1) relative to the starting square, and also to a(14) = 5 at (0,2) relative to the starting square. This is the first term to differ from A366354.
%Y A366353 Cf. A366354, A214526.
%K A366353 nonn
%O A366353 0,5
%A A366353 _Scott R. Shannon_, Oct 08 2023