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List of X-coordinates of point moving along one of the arms of a counterclockwise square spiral with four arms; A340945 gives Y-coordinates.
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%I #19 May 24 2023 06:05:18

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

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

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

%N List of X-coordinates of point moving along one of the arms of a counterclockwise square spiral with four arms; A340945 gives Y-coordinates.

%H Rémy Sigrist, <a href="/A340944/b340944.txt">Table of n, a(n) for n = 0..10101</a>

%H Rémy Sigrist, <a href="/A340944/a340944.gp.txt">PARI program for A340944</a>

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

%F a(n) = A340945(n) iff n belongs to A002939.

%F a(n) = - A340945(n) iff n = 4*x^2 - 6*x + 2 for some x.

%F a(n) = 0 iff n is an even square (A016742).

%F a(n) = Re z where z = (r-s*i)*i^s with s = round(sqrt(n)) = A000194(n) and remainder r = n - s^2. - _Kevin Ryde_, May 24 2023

%e The spiral starts as follows:

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

%e |7 6 5 4 |3 .

%e | | .

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

%e |8 | | |2 |

%e | | | | |

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

%e |9 | | |0 1 |

%e | | | | | .

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

%e |10 | | .

%e | | | .

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

%e .11 .

%e .

%e - so a(0) = a(4) = 0,

%e a(1) = a(2) = a(3) = 1,

%e a(5) = -1,

%e a(6) = -2,

%e a(7) = a(8) = a(9) = a(10) = a(11) = -3.

%o (PARI) See Links section.

%o (PARI) a(n) = my(r,s=sqrtint(n,&r)); if(r>s, s++;r-=2*s-1); real((r-s*I)*I^s); \\ _Kevin Ryde_, May 24 2023

%Y Cf. A000194, A002939, A016742, A340171, A340945.

%K sign,easy,look

%O 0,7

%A _Rémy Sigrist_, Jan 30 2021