A091450 Transpose of A091449: array T(n,k) read by antidiagonals, where column k is the increasing sequence of numbers m for which the simple continued fraction of sqrt(m) has period k, n >= 1, k >= 0.

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

Offset: 1

Author

Clark Kimberling, Feb 03 2004

Keywords

• nonn
• tabl

Comments

A permutation of the positive integers.

Examples

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

Crossrefs

Cf. A002522, A013642, A091449, A091451, A091453.

Extensions

a(19) = T(3,3) corrected by Pontus von Brömssen, Nov 23 2024