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

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.

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.

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.

A290650 a(1) = 1. For n > 1, a(n) = a(n-1)/2 if a(n-1) is even, a(n) = a(n-1)*n otherwise.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 56, 28, 14, 7, 84, 42, 21, 315, 5040, 2520, 1260, 630, 315, 6615, 145530, 72765, 1746360, 873180, 436590, 218295, 6112260, 3056130, 1528065, 47370015, 1515840480, 757920240, 378960120

Offset: 1

John Bailey, Aug 08 2017

nonn

Conjecture: The sequence is unbounded. - Felix Fröhlich, Aug 08 2017

Harvey P. Dale, Table of n, a(n) for n = 1..1000

nxt[{n_,a_}]:={n+1,If[EvenQ[a],a/2,a(n+1)]}; NestList[nxt,{1,1},40][[All,2]] (* Harvey P. Dale, Dec 23 2020 *)

terms(n) = my(x=1, k=1, i=0); while(1, if(i==n, break, if(i==0, print1(x, ", "); k++; i++, if(x%2==0, x=x/2; k++, x=x*k; k++); print1(x, ", "); i++)))
/* Print initial 40 terms as follows */
terms(40) \\ Felix Fröhlich, Aug 08 2017

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User: John Bailey

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

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A337311 List of successive x-coordinates in the acute angled 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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A290650 a(1) = 1. For n > 1, a(n) = a(n-1)/2 if a(n-1) is even, a(n) = a(n-1)*n otherwise.

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