login
a(n) = squared distance to the origin of the n-th vertex on a counterclockwise undulating spiral in a square grid.
4

%I #47 Apr 01 2023 11:22:24

%S 0,1,2,1,4,5,2,5,4,1,2,1,4,5,2,5,4,9,10,5,8,5,10,9,16,17,10,13,8,13,

%T 10,17,16,9,10,5,8,5,10,9,16,17,10,13,8,13,10,17,16,25,26,17,20,13,18,

%U 13,20,17,26,25,36,37,26,29,20,25,18,25,20,29,26,37,36,25,26,17,20,13,18,13,20

%N a(n) = squared distance to the origin of the n-th vertex on a counterclockwise undulating spiral in a square grid.

%C The spiral coordinates are A359216 and A359217.

%H Rémy Sigrist, <a href="/A359058/b359058.txt">Table of n, a(n) for n = 0..10081</a>

%H Hans G. Oberlack, <a href="/A359058/a359058_1.pdf">Undulating counterclockwise spiral in a square grid</a>

%H Rémy Sigrist, <a href="/A359058/a359058.gp.txt">PARI program</a>

%F a(n) = A359216(n)^2 + A359217(n)^2.

%e The spiral begins as follows and for instance point n=7 is at x=-2,y=1 so that a(7) = (-2)^2 + 1^2 = 5.

%e y ^

%e |

%e 4 | 17--16

%e | | |

%e 3 | 13--10 9--10

%e | | |

%e 2 | 13---8 5---4 5---8

%e | | | | |

%e 1 | 17--10 5---2 1---2 5--10

%e | | | | |

%e 0 | 16---9 4---1 0---1 4---9

%e | | | | |

%e -1 | 10---5 2---1 2---5 10--17

%e | | | | |

%e -2 | 8---5 4---5 8--13

%e | | |

%e -3 | 10---9 10--13

%e | | |

%e -4 | 16--17

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

%e -4 -3 -2 -1 0 1 2 3 4 x

%o (PARI) See Links section.

%Y Cf. A001481, A336336, A359216, A359217.

%K nonn,easy

%O 0,3

%A _Hans G. Oberlack_, Dec 14 2022