A297347 -id:A297347 - cp's OEIS frontend

A256111 a(n) = squared distance to the origin of the n-th vertex on a Babylonian Spiral.

0, 1, 5, 13, 26, 50, 65, 85, 116, 100, 97, 85, 85, 90, 128, 205, 293, 409, 481, 586, 730, 845, 890, 841, 833, 745, 514, 244, 65, 17, 106, 338, 698, 1117, 1225, 1193, 1040, 986, 1037, 1060, 850, 477, 197, 85, 80, 232, 530, 757, 650, 522, 225, 16, 50, 333, 797

Offset: 0

Gordon Hamilton, Mar 14 2015

nonn

A Babylonian spiral is constructed by starting with a zero vector and progressively concatenating the next longest vector with integral endpoints on a Cartesian grid.  (The squares of the lengths of these vectors are A001481.)  The direction of the new vector is chosen to create the clockwise spiral which minimizes the change in direction from the previous vector.
     . . . . . . . . . . . . . . . . . . . . . .
     . . . . . . . . . . . . . . . . . . . . . .
     . . . 14. . . . . . . . . . . . . . . . . .
     . . . . . . . . . . . . . . . . . . . . . .
     . . . . . . . . . . . . . . . . . . . . . .
     . . . . . . . . . . . . . . . . . . . . . .
     . . . . . . . . . . . . . . . . . . . . . .
     . . 13. . . . . . . . . . . . . . . . . . .
     . . . . . . . . . . . . 2 . 3 . . . . . . .
     . . . . . . . . . . . 1 . . . . 4 . . . . .
     . . . . . . . . . . . o . . . . . . . . . .
     . . . . . . . . . . . . . . . . . . 5 . . .
     . . 12. . . . . . . . . . . . . . . . . . .
     . . . . . . . . . . . . . . . . . . . . . .
     . . . . . . . . . . . . . . . . . . 6 . . .
     . . . . . . . . . . . . . . . . . . . . . .
     . . . . 11. . . . . . . . . . . . . . . . .
     . . . . . . . . . . . . . . . . . 7 . . . .
     . . . . . . . . . . . . . . . . . . . . . .
     . . . . . . . 10. . . . . . . . . . . . . .
     . . . . . . . . . . . 9 . . . 8 . . . . . .
     . . . . . . . . . . . . . . . . . . . . . .
The name is chosen to mislead school students into making an incorrect hypothesis about the Babylonian Spiral's long-term behavior.

			On the above diagram, point 4 is distance sqrt(26) from the origin, so a(4) = 26.

Lars Blomberg, Table of n, a(n) for n = 0..10000
Lars Blomberg, Illustrations of 100, 1000 and 10000 terms
MathPickle, Babylonian Spiral

x-coordinates given in A297346. y-coordinates given in A297347.

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]]
]
Norm[#]^2 & /@ Accumulate[NestList[NextVec, {0, 1}, 50]] (* Alex Meiburg, Dec 29 2017 *)

Corrected a(16) and more terms from Lars Blomberg, Nov 17 2016

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

Original entry on oeis.org

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

Alex Meiburg, Dec 28 2017

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

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

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.

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

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]]

cp's OEIS Frontend

A256111 a(n) = squared distance to the origin of the n-th vertex on a Babylonian Spiral.

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