cp's OEIS Frontend

A031405 Numbers k such that the continued fraction for sqrt(k) has even period 2*m and the m-th term of the periodic part is 2.

%I A031405 #24 Jul 13 2021 23:42:20
%S A031405 6,12,14,20,21,28,30,33,42,44,45,52,55,56,60,70,72,77,90,95,110,112,
%T A031405 117,126,132,133,138,153,154,156,161,165,180,182,184,189,190,207,209,
%U A031405 210,221,234,240,248,253,261,272,275,276,285,286,297,299,306,310,315
%N A031405 Numbers k such that the continued fraction for sqrt(k) has even period 2*m and the m-th term of the periodic part is 2.
%H A031405 T. D. Noe, <a href="/A031405/b031405.txt">Table of n, a(n) for n = 1..1000</a>
%e A031405 The c.f. for sqrt(6) is [2; 2, 4, ...] with period 2 and 1st term of the periodic part 2.
%e A031405 The c.f. for sqrt(14) is [3; 1, 2, 1, 6, ...] with period 4 and 2nd term of the periodic part 2.
%e A031405 The c.f. for sqrt(21) is [4; 1, 1, 2, 1, 1, 8, ...] with period 6 and 3rd term of the periodic part 2.
%p A031405 filter:= proc(n) local P,l;
%p A031405   if issqr(n) then return false fi;
%p A031405   P:= numtheory:-cfrac(sqrt(n),'periodic','quotients')[2];
%p A031405   l:= nops(P);
%p A031405   if l::odd then false
%p A031405   else P[l/2] = 2
%p A031405   fi
%p A031405 end proc:
%p A031405 select(filter, [$1..1000]); # _Robert Israel_, Apr 14 2016
%t A031405 n = 1; t = {}; While[Length[t] < 50, n++; If[! IntegerQ[Sqrt[n]], c = ContinuedFraction[Sqrt[n]]; len = Length[c[[2]]]; If[EvenQ[len] && c[[2, len/2]] == 2, AppendTo[t, n]]]]; t
%Y A031405 Cf. A031404-A031423.
%K A031405 nonn
%O A031405 1,1
%A A031405 _David W. Wilson_