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%I #17 May 28 2023 11:25:57
%S 0,0,1,2,2,2,2,2,1,0,-1,-2,-3,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-2,-1,0,1,
%T 2,3,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,5,4,3,2,1,0,-1,-2,-3,-4,-5,-6,-7,
%U -8,-8,-8,-8,-8,-8,-8,-8,-8,-8,-8,-8,-8,-8,-8,-8,-8
%N List of Y-coordinates of point moving along one of the arms of a counterclockwise square spiral with four arms; A340944 gives X-coordinates.
%H Rémy Sigrist, <a href="/A340945/b340945.txt">Table of n, a(n) for n = 0..10101</a>
%H Rémy Sigrist, <a href="/A340945/a340945.gp.txt">PARI program for A340945</a>
%H <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F a(n) = 0 iff n = 4*x^2 + 4*x + 1 for some x.
%F a(n) = Im z where z = (r-s*i)*i^s with s = round(sqrt(n)) = A000194(n) and remainder r = n - s^2. - _Kevin Ryde_, May 28 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(1) = a(9) = 0,
%e a(2) = a(8) = 1,
%e a(3) = a(4) = a(5) = a(6) = a(7) = 2,
%e a(10) = -1,
%e a(11) = -2.
%o (PARI) See Links section.
%o (PARI) a(n) = my(r,s=sqrtint(n,&r)); if(r>s, s++;r-=2*s-1); imag((r-s*I)*I^s); \\ _Kevin Ryde_, May 28 2023
%Y Cf. A340172, A340944, A000194.
%K sign,easy,look
%O 0,4
%A _Rémy Sigrist_, Jan 30 2021
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