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a(0) = 0; for n > 0, a(n) is the largest taxicab distance on a square spiral between any two previous occurrences of a(n-1). If a(n-1) has not previously occurred then a(n) = 0.
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%I #12 Oct 16 2023 13:42:50

%S 0,0,1,0,2,0,2,2,3,0,4,0,4,2,5,0,6,0,6,2,6,4,7,0,6,8,0,7,5,4,8,5,4,8,

%T 8,9,0,10,0,10,2,7,8,12,0,10,8,12,4,8,12,7,8,12,10,9,6,12,12,12,12,12,

%U 12,12,13,0,11,0,11,2,8,12,14,0,11,8,12,14,5,8,12,15,0,15,2,9,10,9,10,11

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

%H Scott R. Shannon, <a href="/A366354/b366354.txt">Table of n, a(n) for n = 0..10000</a>

%H Scott R. Shannon, <a href="/A366354/a366354_1.png">Image of the first 500000 terms</a>.

%H Scott R. Shannon, <a href="/A366354/a366354.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 The spiral begins:

%e .

%e .

%e 0---9---8---8---4---5---8 :

%e | | :

%e 10 6---0---5---2---4 4 10

%e | | | | |

%e 0 0 2---0---1 0 5 12

%e | | | | | | |

%e 10 6 0 0---0 4 7 8

%e | | | | | |

%e 2 2 2---2---3---0 0 7

%e | | | |

%e 7 6---4---7---0---6---8 12

%e | |

%e 8--12---0--10---8--12---4---8

%e .

%e a(2) = 1 as the maximum taxicab distance between 0 = a(1) and the only previous occurrence of 0, a(0) at (0,0), is 1.

%e a(8) = 3 as the maximum taxicab distance between any two previous occurrences of 2 = a(7) is 3, between a(3) = 2, at (-1,1) relative to the starting square, and a(7) = 2 at (0,-1) relative to the starting square.

%e a(32) = 4 as the maximum taxicab distance between any two previous occurrences of 5 = a(31) is 4, between a(14) = 5, at (0,2) relative to the starting square, and a(28) = 5 at (3,1) relative to the starting square. This is the first term to differ from A366353.

%Y Cf. A366353, A214526.

%K nonn

%O 0,5

%A _Scott R. Shannon_, Oct 08 2023