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

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, 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, -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

Offset: 1

Author

Alex Meiburg, Dec 28 2017

Keywords

• easy
• sign
• look

Comments

The "Babylonian Spiral" is defined and illustrated in A256111. See also the MathPickle link.

Examples

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

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
• Hugo Pfoertner, Illustration of spiral using Plot 2.

Crossrefs

The y-coordinates are given in A297347. 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]][[;; , 1]]

Extensions

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