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