A297347 - OEIS

%I #10 Dec 28 2017 20:17:23

%S 0,1,2,2,1,-1,-4,-7,-10,-10,-9,-6,-2,3,8,13,17,20,20,19,17,13,7,0,-7,

%T -13,-15,-12,-7,1,9,17,23,26,21,13,4,-5,-14,-22,-25,-21,-14,-6,4,14,

%U 23,26,19,9,0,0,1,3,11,20,30,41,53,65,76,78,71,59,48,44,53,63,75,88,101

%N List of y-coordinates in the Babylonian Spiral.

%C The "Babylonian Spiral" is defined and illustrated in A256111.

%H Alex Meiburg, <a href="/A297347/b297347.txt">Table of n, a(n) for n = 1..10000</a>

%H Lars Blomberg, Illustrations of <a href="/A256111/a256111.png">100</a>, <a href="/A256111/a256111_1.png">1000</a> and <a href="/A256111/a256111_2.png">10000</a> terms of the spiral.

%H MathPickle, <a href="http://mathpickle.com/project/babylonian-spiral">Babylonian Spiral</a>

%e The first few points are (0,0), (0,1), (1,2), (3,2) -- thus the sequence starts out 0, 1, 2, 2.

%t NextVec[{x_, y_}] :=

%t Block[{n = x^2 + y^2 + 1}, While[SquaresR[2, n] == 0, n++];

%t   TakeSmallestBy[

%t      Union[Flatten[(Transpose[

%t         Transpose[Tuples[{1, -1}, 2]] #] & /@

%t         ({{#[[1]], #[[2]]}, {#[[2]], #[[1]]}})) & /@

%t      PowersRepresentations[n, 2, 2], 2]],

%t   Mod[ArcTan[#[[2]], #[[1]]] - ArcTan[y, x], 2 Pi] &, 1][[1]]

%t ]

%t Accumulate[NestList[NextVec, {0, 1}, 500]][[;; , 2]]

%Y The x-coordinates are given in A297346. Norms of vectors are given in A256111.

%K easy,sign,look

%O 1,3

%A _Alex Meiburg_, Dec 28 2017