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 A256111 a(n) = squared distance to the origin of the n-th vertex on a Babylonian Spiral. 6
 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 (list; graph; refs; listen; history; text; internal format)
 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.      . . . . . . . . . . . . . . . . . . . . . .      . . . . . . . . . . . . . . . . . . . . . .      . . . 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. 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

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Last modified April 22 07:26 EDT 2021. Contains 343163 sequences. (Running on oeis4.)