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List of y-coordinates of a point moving in a smooth counterclockwise spiral rotated by Pi/4.
6

%I #34 Jul 30 2021 08:34:53

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

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

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

%N List of y-coordinates of a point moving in a smooth counterclockwise spiral rotated by Pi/4.

%H Hugo Pfoertner, <a href="/A305258/a305258.png">Illustration of spiral</a>.

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

%F a(n) = A053616(n)*sign(sin(Pi*(1+sqrt(1+8*n))/2)), so that abs(a(n)) = A053616(n).

%F a(n) = A010751(n-floor((1/2)*(sqrt(2n-1)+1))). - _William McCarty_, Jul 29 2021

%e Sequence gives y-coordinate of the n-th point of the following spiral:

%e d:

%e 4 | 32 49

%e | / \ \

%e 3 | 33 18 31 48

%e | / / \ \ \

%e 2 | 34 19 8 17 30 47

%e | / / / \ \ \ \

%e 1 | 35 20 9 2 7 16 29 46

%e | / / / / \ \ \ \ \

%e 0 | 36 21 10 3 0---1 6 15 28 45

%e | \ \ \ \ / / / /

%e -1 | 37 22 11 4---5 14 27 44

%e | \ \ \ / / /

%e -2 | 38 23 12--13 26 43

%e | \ \ / /

%e -3 | 39 24--25 42

%e | \ /

%e -4 | 40--41

%e _______________________________________

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

%o (PARI) up=-1;print1(x=0,", ");for(stride=1,12,up=-up;x+=stride;y=x+stride+1;for(k=x,y-1,print1(up*min(k-x,y-k), ", "))) \\ _Hugo Pfoertner_, Jun 02 2018

%Y A010751 gives sequence of x-coordinates.

%Y Cf. A053616.

%Y Cf. A000384, A001105, A002024, A014105, A046092.

%K sign

%O 0,9

%A _Hugo Pfoertner_, May 29 2018