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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.
2
6, 12, 14, 20, 21, 28, 30, 33, 42, 44, 45, 52, 55, 56, 60, 70, 72, 77, 90, 95, 110, 112, 117, 126, 132, 133, 138, 153, 154, 156, 161, 165, 180, 182, 184, 189, 190, 207, 209, 210, 221, 234, 240, 248, 253, 261, 272, 275, 276, 285, 286, 297, 299, 306, 310, 315
OFFSET
1,1
EXAMPLE
The c.f. for sqrt(6) is [2; 2, 4, ...] with period 2 and 1st term of the periodic part 2.
The c.f. for sqrt(14) is [3; 1, 2, 1, 6, ...] with period 4 and 2nd term of the periodic part 2.
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.
MAPLE
filter:= proc(n) local P, l;
if issqr(n) then return false fi;
P:= numtheory:-cfrac(sqrt(n), 'periodic', 'quotients')[2];
l:= nops(P);
if l::odd then false
else P[l/2] = 2
fi
end proc:
select(filter, [$1..1000]); # Robert Israel, Apr 14 2016
MATHEMATICA
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
CROSSREFS
KEYWORD
nonn
STATUS
approved