login
List of X-coordinates of point moving along one of the arms of a counterclockwise double square spiral; A340172 gives Y-coordinates.
3

%I #17 Jan 03 2021 06:43:10

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

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

%U -2,-3,-4,-5,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5

%N List of X-coordinates of point moving along one of the arms of a counterclockwise double square spiral; A340172 gives Y-coordinates.

%C The odd function f such that f(n) = (a(n), A340172(n)) for any n >= 0 will visit exactly once every lattice point (so it is a bijection from Z to Z^2).

%H Rémy Sigrist, <a href="/A340171/b340171.txt">Table of n, a(n) for n = 0..10000</a>

%H Rémy Sigrist, <a href="/A340171/a340171.gp.txt">PARI program for A340171</a>

%H <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>

%F abs(a(n+1)-a(n)) + abs(A340172(n+1)-A340172(n)) = 1.

%F a(n) = A340172(n) iff n belongs to A001105.

%F a(n) = -A340172(n) iff n belongs to A046092.

%F a(n) = 2*A340172(n) iff n belongs to A139274.

%F 2*a(n) = A340172(n) iff n belongs to A139275.

%F a(n) * A340172(n) = 0 iff n belongs to A000217.

%F a(n) = 0 iff n belongs to A014105.

%e The spiral starts as follows:

%e +-----+-----+-----+-----+-----+

%e . |

%e . |

%e . +-----+-----+-----+ +

%e . |5 4 3 |2 |

%e . | | |

%e + +-----+-----+ +

%e |6 | 0 1 | .

%e | | | .

%e + +-----+-----+-----+ .

%e |7 .

%e | .

%e +-----+-----+-----+-----+-----+

%e 8 9 10 11 12 13

%e - so a(0) = a(3) = a(10) = 0,

%e - a(1) = a(2) = a(11) = 1.

%o (PARI) See Links section.

%Y Cf. A000217, A014105, A046092, A001105, A139274, A139275, A174344, A340172.

%K sign

%O 0,6

%A _Rémy Sigrist_, Dec 30 2020