A091450 -id:A091450 - cp's OEIS frontend

A091449 Array T(n,k) read by antidiagonals, where row n is the increasing sequence of numbers m for which the simple continued fraction of sqrt(m) has period n, n >= 0, k >= 1.

1, 2, 4, 3, 5, 9, 41, 6, 10, 16, 7, 130, 8, 17, 25, 13, 14, 269, 11, 26, 36, 19, 29, 23, 370, 12, 37, 49, 58, 21, 53, 28, 458, 15, 50, 64, 31, 73, 22, 74, 32, 697, 18, 65, 81, 106, 44, 202, 45, 85, 33, 986, 20, 82, 100, 43, 113, 69, 250, 52, 89, 34, 1313, 24, 101, 121

Offset: 0

Clark Kimberling, Feb 03 2004

A permutation of the positive integers.

			Array begins:
  n\k|   1   2   3   4   5   6   7    8    9   10   11
  ---+------------------------------------------------
   0 |   1   4   9  16  25  36  49   64   81  100  121
   1 |   2   5  10  17  26  37  50   65   82  101  122
   2 |   3   6   8  11  12  15  18   20   24   27   30
   3 |  41 130 269 370 458 697 986 1313 1325 1613 1714
   4 |   7  14  23  28  32  33  34   47   55   60   62
   5 |  13  29  53  74  85  89 125  173  185  218  229
   6 |  19  21  22  45  52  54  57   59   70   77   88
   7 |  58  73 202 250 274 314 349  425  538  761 1010
   8 |  31  44  69  71  91  92 108  135  153  158  160
   9 | 106 113 137 149 265 389 493  610  698  754  970
  10 |  43  67  86  93 115 116 118  129  154  159  161
The least n for which CF(sqrt(n)) has period of length 4 is n=7, with CF=[2;1,1,1,4,1,1,1,4,1,1,1,4,...]; thus T(4,1)=7.
[The array T(n,k) is indexed by n=0,1,2,3,..., k=1,2,3... .]
Row 0 consists of squares: 1,4,9,...

Cf. A002522, A003285, A013642, A091450, A091451, A091453.

Rows 0-100 are: A000290 (except the initial 0), A002522 (except the initial 1), A013642, A013643, A013644, A010337, A020347, A010338, A020348, A010339, A020349-A020439.

a(17) = T(3,3) corrected by Pontus von Brömssen, Nov 23 2024

A091452 Transpose of array in A091451 associated with period lengths of square roots of positive integers.

Original entry on oeis.org

1, 4, 2, 9, 5, 3, 16, 10, 6, 7, 25, 17, 8, 14, 13, 36, 26, 11, 23, 29, 19, 49, 37, 12, 28, 53, 21, 31, 64, 50, 15, 32, 74, 22, 44, 41, 81, 65, 18, 33, 85, 45, 69, 130, 43, 100, 82, 20, 34, 89, 52, 71, 268, 67, 46

Offset: 1

Clark Kimberling, Feb 03 2004

A permutation of the positive integers.

			Northwest corner of array:
1 2 3 7
4 5 6 14
9 10 8 23
16 17 11 28

Cf. A091450, A091453.

cp's OEIS Frontend

A091449 Array T(n,k) read by antidiagonals, where row n is the increasing sequence of numbers m for which the simple continued fraction of sqrt(m) has period n, n >= 0, k >= 1.

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