A031551 - OEIS

%I #25 Apr 12 2023 11:31:49

%S 2811,2819,2839,2843,2851,2859,2863,2879,2887,2903,2911,2927,2931,

%T 2939,2963,2971,2979,2999,3007,3011,3019,3023,11240,11264,11296,11328,

%U 11336,11424,11432,11456,11520,11560,11584,11616,11624,11648,11680,11712,11720

%N 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.

%C "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

%C 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

%H Robert Israel, <a href="/A031551/b031551.txt">Table of n, a(n) for n = 1..107</a>

%t 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 *)

%K nonn

%O 1,1

%A _David W. Wilson_

%E Definition clarified by _Harvey P. Dale_, Apr 11 2022