# Sequences based on a spiral of the natural numbers

## Frank Ellermann

```A clockwise spiral of the natural numbers starting at 1 (or 0)
in any direction (here: south-north) can be used to visualize
many simple sequences:

82  83  84  85  86  87  88  89  90  91
81  50  51  52  53  54  55  56  57  92
80  49  26  27  28  29  30  31  58  93
79  48  25  10  11  12  13  32  59  94
78  47  24  9   2   3   14  33  60  95
77  46  23  8   1   4   15  34  61  96
76  45  22  7   6   5   16  35  62  97
75  44  21  20  19  18  17  36  63  98
74  43  42  41  40  39  38  37  64  99
73  72  71  70  69  68  67  66  65 100

Rook   moves 1- 2-11 etc.: EIS A054552, 4*n*n -11*n +8, A001107(n)+1
Bishop moves 1- 3-13 etc.: EIS A054554, 4*n*n -10*n +7, A002939(n)+1
Rook   moves 1- 4-15 etc.: EIS A054556, 4*n*n - 9*n +6, A033991(n)+1
Bishop moves 1- 5-17 etc.: EIS A053755, 4*n*n - 8*n +5, A016742(n)+1
Rook   moves 1- 6-19 etc.: EIS A054567, 4*n*n - 7*n +4, A007742(n)+1
Bishop moves 1- 7-21 etc.: EIS A054569, 4*n*n - 6*n +3, A002943(n)+1
Rook   moves 1- 8-23 etc.: EIS A033951, 4*n*n - 5*n +2, A033954(n)+1
Bishop moves 1- 9-25 etc.: EIS A016754, 4*n*n - 4*n +1, A033996(n)+1

Image of spiral shifted in 1-2 direction (y+1): A068225
Image of spiral shifted in 1-6 direction (y-1): A068226
Image of spiral shifted in any direction: more permutations.

There a many more EIS sequences based on this or similar spirals:

A053999, A054551, A054553, A054555, A054564, A054566 (primes),
A033952, A033953, A033988, A033989, A033990 (digits), etc.

It's not really necessary to submit ALL of these sequences... :-)

Rook   moves 1-11-53 etc.: not listed , 16*n*n-38*n+23
Bishop moves 1-13-57 etc.: not listed , 16*n*n-36*n+21
Rook   moves 1-15-61 etc.: not listed , 16*n*n-34*n+19
Bishop moves 1-17-65 etc.: not listed , 16*n*n-32*n+17, A016802(n)+1
Rook   moves 1-19-69 etc.: not listed , 16*n*n-30*n+15
Bishop moves 1-21-73 etc.: not listed , 16*n*n-28*n+13
Rook   moves 1-23-77 etc.: not listed , 16*n*n-26*n+11
Bishop moves 1-25-81 etc.: EIS A016814, 16*n*n-24*n+ 9

Knight moves 1-10-51 etc.: not listed , 16*n*n-39*n+24
Knight moves 1-12-55 etc.: not listed , 16*n*n-37*n+22
Knight moves 1-14-59 etc.: not listed , 16*n*n-35*n+20
Knight moves 1-16-63 etc.: not listed , 16*n*n-33*n+18
Knight moves 1-18-67 etc.: not listed , 16*n*n-31*n+16
Knight moves 1-20-71 etc.: not listed , 16*n*n-29*n+14
Knight moves 1-22-75 etc.: not listed , 16*n*n-27*n+12
Knight moves 1-24-79 etc.: not listed , 16*n*n-25*n+10
```

Last update: 21 Mar 2002 00:50 by F.   Ellermann