A031551 Numbers k such that the continued fraction for sqrt(k) has even period and if the last term of the periodic part is deleted the central term is 53.

2811, 2819, 2839, 2843, 2851, 2859, 2863, 2879, 2887, 2903, 2911, 2927, 2931, 2939, 2963, 2971, 2979, 2999, 3007, 3011, 3019, 3023, 11240, 11264, 11296, 11328, 11336, 11424, 11432, 11456, 11520, 11560, 11584, 11616, 11624, 11648, 11680, 11712, 11720

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

"Central term" means the term at 1/2 of the length of the repeating part, not the term following that term, e.g., if the terms are {a,b,c,d}, the "central term" is b, not c. - Harvey P. Dale, May 19 2012

Includes 2809 * k^2 + 2 * k for k >= 1, where the continued fraction has initial term 53*k and periodic part [53, 106*k], and 3025 * k^2 - 2 * k for k >= 1, where the continued fraction has initial term 55*k-1 and periodic part [1, 53, 1, 110*k-2]. - Robert Israel, Apr 11 2023

Links

• Robert Israel, Table of n, a(n) for n = 1..107

Programs

Mathematica

epQ[n_]:=Module[{p=ContinuedFraction[Sqrt[n]][[2]],len},len=Length[p];EvenQ[len]&&p[[len/2]]==53]; nn=12000;With[{trms=Complement[Range[ nn], Range[Floor[Sqrt[nn]]]^2]},Select[trms,epQ]]  (* Harvey P. Dale, May 19 2012 *)

Extensions

Definition clarified by Harvey P. Dale, Apr 11 2022