A340945 - OEIS

%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