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Sum of x- and y-coordinate of a point moving in a triangular spiral.
0

%I #11 May 27 2021 11:14:39

%S 0,1,1,-1,-3,-2,-1,0,1,2,2,2,2,0,-2,-4,-6,-5,-4,-3,-2,-1,0,1,2,3,3,3,

%T 3,3,3,1,-1,-3,-5,-7,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,4,4,4,4,4,4,

%U 4,2,0,-2,-4,-6,-8,-10,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,5,5,5,5,5,5,5,5,5,3,1,-1,-3,-5,-7,-9,-11,-13,-15

%N Sum of x- and y-coordinate of a point moving in a triangular spiral.

%e y

%e |

%e 4 | 56

%e | \

%e | \

%e | \

%e 3 | 30 55

%e | / \ \

%e | / \ \

%e | / \ \

%e 2 | 31 12 29 54

%e | / / \ \ \

%e | / / \ \ \

%e | / / \ \ \

%e 1 | 32 13 2 11 28 53

%e | / / / \ \ \ \

%e | / / / \ \ \ \

%e | / / / \ \ \ \

%e 0 | 33 14 3 0---1 10 27 52

%e | / / / \ \ \

%e | / / / \ \ \

%e | / / / \ \ \

%e -1 | 34 15 4---5---6---7---8---9 26 51

%e | / / \ \

%e | / / \ \

%e | / / \ \

%e -2 | 35 16--17--18--19--20--21--22--23--24--25 50

%e | / \

%e | / \

%e | / \

%e -3 | 36--37--38--39--40--41--42--43--44--45--46--47--48--49

%e |

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

%e x: -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

%e Count as follows. Start at n=0 with coordinates (0,0). So first term is 0.

%e Next step is to move to n=1 with coordinates (1,0) resulting in sum=1. So the sequence begins 0,1,...

%e Next step is to move to n=2 with coordinates (0,1) resulting in sum=1. So the sequence begins 0,1,1,...

%e Next step is to move to n=3 with coordinates (-1,0) resulting in sum=-1. So the sequence begins 0,1,1,1,...

%e Next step is to move to n=4 with coordinates (-2,-1) resulting in sum=-3. So the sequence begins 0,1,1,-1,-3,...

%e and so on.

%Y Cf. A329972, A329116.

%K sign

%O 0,5

%A _Hans G. Oberlack_, Apr 07 2021