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Successively count to (-1)^(n+1)*n (n = 0, 1, 2, ... ).
7

%I #84 Aug 05 2022 07:45:55

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

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

%U -1,-2,-3,-4,-5,-6,-7,-8

%N Successively count to (-1)^(n+1)*n (n = 0, 1, 2, ... ).

%C Also x-coordinates of a point moving in counterclockwise triangular spiral (A329972 gives the y-coordinates).

%H Rémy Sigrist, <a href="/A329116/b329116.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(n) = (-1)^t * (t^2 - t - n) where t=ceiling(sqrt(n)).

%F a(n) = (-1)^t * floor(t^2 - sqrt(n) - n) where t=ceiling(sqrt(n)).

%F A053615(n) = abs(a(n)).

%F abs(A196199(n)) = abs(a(n)).

%F A255175(n) = a(n+1).

%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 We count as follows. Start at n=0 with 0.

%e Next step is to count to 1: so we have 0, 1.

%e Next step is to count to -2, so we have 0, 1, 0, -1, -2.

%e Next we have to go to +3, so we have 0, 1, 0, -1, -2, -1, 0, 1, 2, 3.

%e And so on.

%t a[n_] := Table[(-1)^(# + 1)*(-#^2 + # + k) &[Ceiling@ Sqrt@ k], {k, 0, n}]; a[64]

%o (Python)

%o from math import isqrt

%o def A329116(n): return ((t:=1+isqrt(n-1))*(t-1)-n)*(-1 if t&1 else 1) if n else 0 # _Chai Wah Wu_, Aug 04 2022

%Y Cf. A053615, A196199, A339265 (first differences). Essentially the same as A255175.

%K sign,easy,look

%O 0,5

%A _Mikk Heidemaa_, Nov 13 2019