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.

%I A091449 #11 Nov 24 2024 09:25:41
%S A091449 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,
%T A091449 12,37,49,58,21,53,28,458,15,50,64,31,73,22,74,32,697,18,65,81,106,44,
%U A091449 202,45,85,33,986,20,82,100,43,113,69,250,52,89,34,1313,24,101,121
%N 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.
%C A091449 A permutation of the positive integers.
%e A091449 Array begins:
%e A091449   n\k|   1   2   3   4   5   6   7    8    9   10   11
%e A091449   ---+------------------------------------------------
%e A091449    0 |   1   4   9  16  25  36  49   64   81  100  121
%e A091449    1 |   2   5  10  17  26  37  50   65   82  101  122
%e A091449    2 |   3   6   8  11  12  15  18   20   24   27   30
%e A091449    3 |  41 130 269 370 458 697 986 1313 1325 1613 1714
%e A091449    4 |   7  14  23  28  32  33  34   47   55   60   62
%e A091449    5 |  13  29  53  74  85  89 125  173  185  218  229
%e A091449    6 |  19  21  22  45  52  54  57   59   70   77   88
%e A091449    7 |  58  73 202 250 274 314 349  425  538  761 1010
%e A091449    8 |  31  44  69  71  91  92 108  135  153  158  160
%e A091449    9 | 106 113 137 149 265 389 493  610  698  754  970
%e A091449   10 |  43  67  86  93 115 116 118  129  154  159  161
%e A091449 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.
%e A091449 [The array T(n,k) is indexed by n=0,1,2,3,..., k=1,2,3... .]
%e A091449 Row 0 consists of squares: 1,4,9,...
%Y A091449 Cf. A002522, A003285, A013642, A091450, A091451, A091453.
%Y A091449 Rows 0-100 are: A000290 (except the initial 0), A002522 (except the initial 1), A013642, A013643, A013644, A010337, A020347, A010338, A020348, A010339, A020349-A020439.
%K A091449 nonn,tabl
%O A091449 0,2
%A A091449 _Clark Kimberling_, Feb 03 2004
%E A091449 a(17) = T(3,3) corrected by _Pontus von Brömssen_, Nov 23 2024