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

1,4

Comments

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

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.

Example

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

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]]

Crossrefs

The y-coordinates are given in A297347. Norms of vectors are given in A256111.

Sequence in context: A092257 A316189 A342668 * A342622 A211517 A254935

Adjacent sequences: A297343 A297344 A297345 * A297347 A297348 A297349

Keyword

easy,sign,look

Author

Alex Meiburg, Dec 28 2017

Extensions

~

Status

approved