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A359217 Y-coordinates of a point moving along a counterclockwise undulating spiral on a square grid.

%I A359217 #46 Apr 01 2023 11:22:49
%S A359217 0,0,1,1,2,2,1,1,0,0,-1,-1,-2,-2,-1,-1,0,0,1,1,2,2,3,3,4,4,3,3,2,2,1,
%T A359217 1,0,0,-1,-1,-2,-2,-3,-3,-4,-4,-3,-3,-2,-2,-1,-1,0,0,1,1,2,2,3,3,4,4,
%U A359217 5,5,6,6,5,5,4,4,3,3,2,2,1,1,0,0,-1,-1
%N A359217 Y-coordinates of a point moving along a counterclockwise undulating spiral on a square grid.
%C A359217 X-coordinates are given in A359216.
%H A359217 Rémy Sigrist, <a href="/A359217/b359217.txt">Table of n, a(n) for n = 0..10081</a>
%H A359217 Hans G. Oberlack, <a href="/A359217/a359217.pdf">Counterclockwise undulating spiral in a square grid</a>
%H A359217 Rémy Sigrist, <a href="/A359217/a359217.gp.txt">PARI program</a>
%H A359217 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A359217 Conjecture: a(n) = T10 + T15 + T20 + T21 where
%F A359217   T1  = floor(n/16);
%F A359217   T2  = sqrt(2*T1 + 1/4);
%F A359217   T3  = floor(T2 - 1/2);
%F A359217   T4  = n - T3*(T3+1)*16/2;
%F A359217   T5  = (T3+1)*16;
%F A359217   T6  = T4 + (3/4)*T5 - 1;
%F A359217   T7  = T6/T5;
%F A359217   T8  = floor(T7);
%F A359217   T9  = 1 - T8;
%F A359217   T10 = T9 - floor(T4/2);
%F A359217   T11 = T4 + (2/4)*T5 - 1;
%F A359217   T12 = T11/T5;
%F A359217   T13 = floor(T12);
%F A359217   T14 = T8 - T13;
%F A359217   T15 = T14*floor((T5 - T11)/2);
%F A359217   T16 = T4 + (1/4)*T5 - 1;
%F A359217   T17 = T16/T5;
%F A359217   T18 = floor(T17);
%F A359217   T19 = T13 - T18;
%F A359217   T20 = -T19*floor((T4 - T5/2)/2);
%F A359217   T21 = -T18*floor((T5 - T4 + 1)/2).
%F A359217 a(2*n) = A180714(n). - _Rémy Sigrist_, Apr 01 2023
%e A359217    y ^
%e A359217      |
%e A359217    4 |             25--24
%e A359217      |              |   |
%e A359217    3 |         27--26  23--22
%e A359217      |          |           |
%e A359217    2 |     29--28   5---4  21--20
%e A359217      |      |       |   |       |
%e A359217    1 | 31--30   7---6   3---2  19--18
%e A359217      |  |       |           |       |
%e A359217    0 | 32--33   8---9   0---1  16--17
%e A359217      |      |       |           |       |
%e A359217   -1 |     34--35  10--11  14--15  46--47
%e A359217      |          |       |   |       |
%e A359217   -2 |         36--37  12--13  44--45
%e A359217      |              |           |
%e A359217   -3 |             38--39  42--43
%e A359217      |                  |   |
%e A359217   -4 |                 40--41
%e A359217      +------------------------------------>
%e A359217        -4  -3  -2  -1   0   1   2   3   4 x
%o A359217 (PARI) See Links section.
%Y A359217 Cf. A180714, A359058, A359216.
%K A359217 sign,walk
%O A359217 0,5
%A A359217 _Hans G. Oberlack_, Dec 21 2022