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

0,2

Comments

A permutation of the positive integers.

Links

Table of n, a(n) for n=0..65.

Example

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,...

Crossrefs

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.

Sequence in context: A214928 A379649 A240277 * A100834 A264996 A100826

Adjacent sequences: A091446 A091447 A091448 * A091450 A091451 A091452

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 03 2004

Extensions

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

Status

approved