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Triangle T read by rows: T(n,k) equals the Manhattan distance between n and k when integers are arranged in a spiral.
0

%I #12 Aug 13 2018 09:12:04

%S 0,1,0,2,1,0,1,2,1,0,2,3,2,1,0,1,2,3,2,1,0,2,3,4,3,2,1,0,1,2,3,2,3,2,

%T 1,0,2,1,2,3,4,3,2,1,0,3,2,3,4,5,4,3,2,1,0,2,1,2,3,4,3,4,3,2,1,0,3,2,

%U 1,2,3

%N Triangle T read by rows: T(n,k) equals the Manhattan distance between n and k when integers are arranged in a spiral.

%C Spiral begins (0 at the center):

%C .

%C 42--43--44--45--46--47--48

%C |

%C 41 20--21--22--23--24--25

%C | | |

%C 40 19 6---7---8---9 26

%C | | | | |

%C 39 18 5 0---1 10 27

%C | | | | | |

%C 38 17 4---3---2 11 28

%C | | | |

%C 37 16--15--14--13--12 29

%C | |

%C 36--35--34--33--32--31--30

%F T(n,n) = 0, T(n+1,n) = 1, T(n+2,n) = 2, T(n,k) <= n-k.

%e Triangle begins, 0 <= k <= n:

%e 0

%e 1, 0

%e 2, 1, 0

%e 1, 2, 1, 0

%e 2, 3, 2, 1, 0

%e 1, 2, 3, 2, 1, 0

%e 2, 3, 4, 3, 2, 1, 0

%e 1, 2, 3, 2, 3, 2, 1, 0

%e 2, 1, 2, 3, 4, 3, 2, 1, 0

%e 3, 2, 3, 4, 5, 4, 3, 2, 1, 0

%e 2, 1, 2, 3, 4, 3, 4, 3, 2, 1, 0

%e 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0

%e 4, 3, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0

%e 3, 2, 1, 2, 3, 4, 5, 4, 3, 4, 3, 2, 1, 0

%e 2, 3, 2, 1, 2, 3, 4, 3, 4, 5, 4, 3, 2, 1, 0

%e 3, 4, 3, 2, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0

%K nonn,tabl

%O 0,4

%A _Philippe Deléham_, Mar 11 2013