A256111 -id:A256111 - cp's OEIS frontend

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

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]]

~

A297347 List of y-coordinates in the Babylonian Spiral.

Original entry on oeis.org

0, 1, 2, 2, 1, -1, -4, -7, -10, -10, -9, -6, -2, 3, 8, 13, 17, 20, 20, 19, 17, 13, 7, 0, -7, -13, -15, -12, -7, 1, 9, 17, 23, 26, 21, 13, 4, -5, -14, -22, -25, -21, -14, -6, 4, 14, 23, 26, 19, 9, 0, 0, 1, 3, 11, 20, 30, 41, 53, 65, 76, 78, 71, 59, 48, 44, 53, 63, 75, 88, 101

Offset: 1

Alex Meiburg, Dec 28 2017

The "Babylonian Spiral" is defined and illustrated in A256111.

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

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

The x-coordinates are given in A297346. 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]][[;; , 2]]

A337293 a(n) is the squared distance to the origin of the n-th vertex on an acute angled Babylonian spiral.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 5, 1, 8, 8, 5, 17, 1, 26, 0, 29, 5, 25, 25, 18, 34, 5, 50, 1, 49, 17, 18, 52, 5, 85, 2, 90, 18, 61, 125, 13, 148, 10, 153, 20, 98, 125, 41, 145, 4, 148, 18, 85, 170, 18, 225, 148, 202, 173, 61, 197, 41, 226, 10, 229, 25, 117, 170, 5, 208, 80

Offset: 0

John Bailey, Aug 21 2020

nonn

An acute angled 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 maximize the change in direction from the previous vector. The Babylonian spiral (A256111) minimizes this angle.

			The coordinates of the first few points are (0,0), (0,1), (1,0), (-1,0), (1,1), (-1,-1), (-1,2).

John Bailey, Table of n, a(n) for n = 0..10000
John Bailey, Illustrations of 10, 100, 1000, 10000, 100000, 1500000, 10000000
John Bailey, Python code with plots

x-coordinates given in A337311. y-coordinates given in A337312.

Cf. A001481, A256111.

Python
```
# See Bailey link.
```

a(n) = A337311(n)^2 + A337312(n)^2.

A337311 List of successive x-coordinates in the acute angled Babylonian spiral.

Original entry on oeis.org

0, 0, 1, -1, 1, -1, -1, 0, -2, 2, -2, 1, -1, -1, 0, -2, 2, -3, 3, -3, 3, -2, 1, 1, 0, 4, -3, 4, -2, -2, -1, -3, 3, -5, 2, -2, -2, -1, -3, 2, -7, 2, -5, 1, 0, -2, 3, -7, 1, -3, -9, 2, -9, 2, -6, 1, -5, -1, -1, -2, 3, -9, 1, -2, -8, 4, -5, 3, -2, -1, 1, -2, 4

Offset: 1

John Bailey, Aug 22 2020

sign

Cf. A337293, A256111, A337312.

A337312 List of successive y-coordinates in the acute angled Babylonian spiral.

Original entry on oeis.org

0, 1, 0, 0, 1, -1, 2, -1, 2, 2, 1, 4, 0, 5, 0, 5, 1, 4, 4, 3, 5, 1, 7, 0, 7, 1, 3, 6, 1, 9, 1, 9, 3, 6, 11, 3, 12, 3, 12, 4, 7, 11, 4, 12, 2, 12, 3, 6, 13, 3, 12, 12, 11, 13, 5, 14, 4, 15, 3, 15, 4, 6, 13, 1, 12, 8, -1, 9, -3, 10, -3, 10, -2, 8, 0, 7, 2, 2, 3

Offset: 1

John Bailey, Aug 22 2020

sign

Cf. A256111, A337293, A337311.

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

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A297347 List of y-coordinates in the Babylonian Spiral.

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A337293 a(n) is the squared distance to the origin of the n-th vertex on an acute angled Babylonian spiral.

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A337311 List of successive x-coordinates in the acute angled Babylonian spiral.

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A337312 List of successive y-coordinates in the acute angled Babylonian spiral.

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