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

0,3

Comments

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.

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. . . 14. . . . . . . . . . . . . . . . . .

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. . 13. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . 2 . 3 . . . . . . .

. . . . . . . . . . . 1 . . . . 4 . . . . .

. . . . . . . . . . . o . . . . . . . . . .

. . . . . . . . . . . . . . . . . . 5 . . .

. . 12. . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . 6 . . .

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. . . . 11. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . 7 . . . .

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. . . . . . . 10. . . . . . . . . . . . . .

. . . . . . . . . . . 9 . . . 8 . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

The name is chosen to mislead school students into making an incorrect hypothesis about the Babylonian Spiral's long-term behavior.

Links

Lars Blomberg, Table of n, a(n) for n = 0..10000

Lars Blomberg, Illustrations of 100, 1000 and 10000 terms

MathPickle, Babylonian Spiral

Example

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

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

Crossrefs

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

Sequence in context: A014813 A180671 A211637 * A322417 A160420 A182840

Adjacent sequences: A256108 A256109 A256110 * A256112 A256113 A256114

Keyword

nonn

Author

Gordon Hamilton, Mar 14 2015

Extensions

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

Status

approved