OEIS/Spirals

Hexagonal grid, counterclockwise

Cf. A056105 First spoke of a hexagonal spiral.

From Robert G. Wilson v, Jul 05 2014: (Start)
Each of the 6 primary spokes or rays has a generating formula as stated here:
1st:  90 degrees A056105 3n^2 - 2n + 1
2nd:  30 degrees A056106 3n^2 -  n + 1
3rd: 330 degrees A056107 3n^2      + 1
4th: 270 degrees A056108 3n^2 +  n + 1
5th: 210 degrees A056109 3n^2 + 2n + 1
6th: 150 degrees A003215 3n^2 + 3n + 1
Each of the 6 secondary spokes or rays has a generating formula as stated here:
1st:  60 degrees 12n^2 - 27n + 16
2nd: 360 degrees 12n^2 - 25n + 14
3rd: 300 degrees 12n^2 - 23n + 12
4th: 240 degrees 12n^2 - 21n + 10
5th: 180 degrees 12n^2 - 19n +  8
6th: 120 degrees 12n^2 - 17n +  6 = A033577(n+1)

Square grid, counterclockwise

Cf. A244677 The spiral of Champernowne, read along the East ray. Robert G. Wilson v, Cover of the March 1964 issue of Scientific American Formulas for rays in directions of 32 compass points:

 SE     4n^2   -4n  +1
 SExS  64n^2 -113n +50
 SSE   16n^2  -25n +10
 SxE   64n^2 -115n +52
 S      4n^2   -5n  +2
 SxW   64n^2 -117n +54
 SSW   16n^2  -27n +12
 SWxS  64n^2 -119n +56
 SW     4n^2   -6n  +3
 SWxW  64n^2 -121n +58
 WSW   16n^2  -29n +14
 WxS   64n^2 -123n +60
 W      4n^2   -7n  +4
 WxN   64n^2 -125n +62
 WNW   16n^2  -31n +16
 NWxW  64n^2 -127n +64
 NW     4n^2   -8n  +5
 NWxN  64n^2 -129n +66
 NNW   16n^2  -33n +18
 NxW   64n^2 -131n +68
 N      4n^2   -9n  +6
 NxE   64n^2 -133n +70
 NNE   16n^2  -35n +20
 NExN  64n^2 -135n +72
 NE     4n^2  -10n  +7
 NExE  64n^2 -137n +74
 ENE   16n^2  -37n +22
 ExN   64n^2 -139n +76
 E      4n^2  -11n  +8
 ExS   64n^2 -141n +78
 ESE   16n^2  -39n +24
 SExE  64n^2 -143n +80