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A309666 a(n) is the least k such that the denominators of continued fraction convergents for sqrt(k) match the first n Fibonacci numbers.

%I A309666 #24 Aug 26 2019 11:19:42
%S A309666 2,3,7,7,13,58,58,135,819,819,2081,13834,13834,35955,244647,244647,
%T A309666 639389,4374866,4374866,11448871,78439683,78439683,205337953,
%U A309666 1407271538,1407271538,3684200835,25251313255,25251313255,66108441037,453111560266,453111560266,1186259960295,8130736409715,8130736409715,21286537898177
%N A309666 a(n) is the least k such that the denominators of continued fraction convergents for sqrt(k) match the first n Fibonacci numbers.
%C A309666 Aside from the first term, this appears to be a subset of A060215.
%C A309666 Same as A071296 if you drop a(0) and replace each repeated pair x,x with 0,x (credit to _Daniel Suteu_ for pointing this out).
%C A309666 These are also the least a(n) such that the continued fraction expansion for sqrt(a(n) - floor(a(n))) begins with (n-1) 1's.
%F A309666 Conjectures from _Colin Barker_, Aug 26 2019: (Start)
%F A309666 G.f.: x*(2 + x + 4*x^2 - 42*x^3 - 15*x^4 - 39*x^5 + 100*x^6 + x^7 - 61*x^8 + 172*x^9 + 31*x^10 - 17*x^11 + 26*x^12 - 2*x^13 + x^14 - 2*x^15) / ((1 - x)*(1 + x)*(1 - 3*x + x^2)*(1 - x + x^2)*(1 - x - x^2)*(1 + x + 2*x^2 - x^3 + x^4)*(1 + 3*x + 8*x^2 + 3*x^3 + x^4)).
%F A309666 a(n) = a(n-1) + 21*a(n-3) - 21*a(n-4) - 50*a(n-6) + 50*a(n-7) - 86*a(n-9) + 86*a(n-10) - 13*a(n-12) + 13*a(n-13) + a(n-15) - a(n-16) for n>16.
%F A309666 (End)
%e A309666 For n = 5 the convergents of sqrt(13) are 3/1, 4/1, 7/2, 11/3, 18/5, 119/33, ... and the first five denominators are 1, 1, 2, 3, 5, which match the first five Fibonacci numbers. Since 13 is the first number with this property, then a(5) = 13.
%t A309666 c = 1;
%t A309666 n = 2;
%t A309666 F = Table[Fibonacci[n], {n, 20}];
%t A309666 While[c <= 14,
%t A309666 If[! IntegerQ[Sqrt[n]]
%t A309666    &&
%t A309666    Denominator[Convergents[Sqrt[n], c]] == F[[1 ;; c]],
%t A309666   Print[n, "  ", Denominator[Convergents[Sqrt[n], c]]];
%t A309666   c++; n--];
%t A309666 n++
%t A309666 ]
%Y A309666 Cf. A060215, A071296.
%K A309666 nonn
%O A309666 1,1
%A A309666 _Greg Dresden_, Aug 11 2019