# Recursively Rotated Possible Permutation

Leroy Quet qq-quet at mindspring.com
Thu Sep 9 15:34:22 CEST 2004

```Hugo Pfoertner wrote:
>Leroy Quet wrote:
>>
>> I just submitted the following sequence to the EIS.
>>
>> %S A000001 1,3,4,2,7,8,6,11,12
>> %N A000001 Start with positive integers. On m_th step shift terms a(m+1)
>> to a(2m-1) one position to the left, move a(m) to position (2m-1).
>> Sequence is limit of reordering.
>> %e A000001 [1,2,3,4,5,6,...]->[1,3,2,4,5,6,...]->[1,3,4,5,2,6,...]
>> ->[1,3,4,2,6,7,...]->[1,3,4,2,7,5,...]->[1,3,4,2,7,8,...]
>> ->...
>> %O A000001 1
>> %K A000001 ,more,nonn,
>>
>> Is it a permutation of the positive integers?
>> (Or do some integers get rotated out to infinity?)
>>
>> I bet there is a simple, even trivial, way to express which integer is at
>> any particular position. (But a maybe not.)
>>
>> Could someone please extend this sequence once it appears, and if it is a
>> permutation, submit the inverse?
>>
>> thanks,
>> Leroy Quet
>
>I get (15 numbers / line, revealing a nice structure):
>
>  1   3   4   2   7   8   6  11  12   9  15  16   5  19  20
> 14  23  24  17  27  28  13  31  32  22  35  36  25  39  40
> 18  43  44  30  47  48  33  51  52  10  55  56  38  59  60
> 41  63  64  29  67  68  46  71  72  49  75  76  34  79  80
> 54  83  84  57  87  88  26  91  92  62  95  96  65  99 100
> 45 103 104  70 107 108  73 111 112  50 115 116  78 119 120
> 81 123 124  37 127 128  86 131 132  89 135 136  61 139 140
> 94 143 144  97 147 148  66 151 152 102 155 156 105 159 160
> 21 163 164 110 167 168 113 171 172  77 175 176 118 179 180
>121 183 184  82 187 188 126 191 192 129 195 196  58 199 200
>
>and the inverse
>
>  1   4   2   3  13   7   5   6  10   40   8   9  22  16  11
> 12  19  31  14  15 121  25  17  18   28  67  20  21  49  34
> 23  24  37  58  26  27  94  43  29   30  46 364  32  33  76
> 52  35  36  55  85  38  39 202  61   41  42  64 148  44  45
>103  70  47  48  73 112  50  51 175   79  53  54  82 283  56
> 57 130  88  59  60  91 139  62  63 1093  97  65  66 100 229
> 68  69 157 106  71  72 109 166  74   75 256 115  77  78 118
>607  80  81 184 124  83  84 127 193   86  87 445 133  89  90
>136 310  92  93 211 142  95  96 145  220  98  99 337 151 101
>102 154 526 104 105 238 160 107 108  163 247 110 111 850 169
>
>Hugo Pfoertner

Hmmm... I wonder now about any patterns when the sequence is taken mod m
(for other m's besides 10).

thanks,
Leroy Quet

```