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A297346 List of successive x-coordinates in the Babylonian Spiral.

%I A297346 #14 Nov 10 2023 08:43:29
%S A297346 0,0,1,3,5,7,7,6,4,0,-4,-7,-9,-9,-8,-6,-2,3,9,15,21,26,29,29,28,24,17,
%T A297346 10,4,4,5,7,13,21,28,32,32,31,29,24,15,6,-1,-7,-8,-6,-1,9,17,21,15,4,
%U A297346 -7,-18,-26,-33,-39,-43,-43,-42,-37,-25,-15,-12,-18,-30,-39,-47,-52,-53,-51,-48,-42,-33,-22,-10,3,17,31,45,56,62,54,40,27
%N A297346 List of successive x-coordinates in the Babylonian Spiral.
%C A297346 The "Babylonian Spiral" is defined and illustrated in A256111. See also the MathPickle link.
%H A297346 Alex Meiburg, <a href="/A297346/b297346.txt">Table of n, a(n) for n = 1..10000</a>
%H A297346 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 A297346 MathPickle, <a href="http://mathpickle.com/project/babylonian-spiral">Babylonian Spiral</a>
%H A297346 Hugo Pfoertner, <a href="/plot2a?name1=A297346&amp;name2=A297347&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=xy&amp;drawlines=true">Illustration of spiral</a> using Plot 2.
%e A297346 The first few points are (0,0), (0,1), (1,2), (3,2) -- thus the sequence starts out 0, 0, 1, 3.
%t A297346 NextVec[{x_, y_}] :=
%t A297346 Block[{n = x^2 + y^2 + 1}, While[SquaresR[2, n] == 0, n++];
%t A297346   TakeSmallestBy[
%t A297346      Union[Flatten[(Transpose[
%t A297346         Transpose[Tuples[{1, -1},2]] #] & /@
%t A297346         ({{#[[1]], #[[2]]}, {#[[2]], #[[1]]}})) & /@
%t A297346      PowersRepresentations[n, 2, 2], 2]],
%t A297346   Mod[ArcTan[#[[2]], #[[1]]] - ArcTan[y, x], 2 Pi] &, 1][[1]]
%t A297346 ]
%t A297346 Accumulate[NestList[NextVec, {0, 1}, 500]][[;; , 1]]
%Y A297346 The y-coordinates are given in A297347. Norms of vectors are given in A256111.
%K A297346 easy,sign,look
%O A297346 1,4
%A A297346 _Alex Meiburg_, Dec 28 2017
%E A297346   ~