A297347 List of y-coordinates in the Babylonian Spiral.

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

Offset: 1

Author

Alex Meiburg, Dec 28 2017

Keywords

• easy
• sign
• look

Comments

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

Examples

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

Links

• Alex Meiburg, Table of n, a(n) for n = 1..10000
• Lars Blomberg, Illustrations of 100, 1000 and 10000 terms of the spiral.
• MathPickle, Babylonian Spiral

Crossrefs

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

Programs

Mathematica

NextVec[{x_, y_}] :=
Block[{n = x^2 + y^2 + 1}, While[SquaresR[2, n] == 0, n++];
  TakeSmallestBy[
     Union[Flatten[(Transpose[
        Transpose[Tuples[{1, -1}, 2]] #] & /@
        ({{#[[1]], #[[2]]}, {#[[2]], #[[1]]}})) & /@
     PowersRepresentations[n, 2, 2], 2]],
  Mod[ArcTan[#[[2]], #[[1]]] - ArcTan[y, x], 2 Pi] &, 1][[1]]
]
Accumulate[NestList[NextVec, {0, 1}, 500]][[;; , 2]]