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A228932 Optimal ascending continued fraction expansion of sqrt(43) - 6.

%I A228932 #40 Dec 27 2016 02:35:27
%S A228932 2,9,30,60,122,-878,11429,35241,-177141,709582,-3123032,-1157723745,
%T A228932 3237738813,-16178936725,33395053634,-71863018424,-153349368674,
%U A228932 -386763022623,-8021033029400,16314606875900,52522689388692
%N A228932 Optimal ascending continued fraction expansion of sqrt(43) - 6.
%C A228932 See A228929 for the definition of "optimal ascending continued fraction".
%C A228932 In A228931 it is shown that many numbers of the type sqrt(x) seem to present in their expansion a recurrence relation a(n) = a(n-1)^2 - 2 between the terms, starting from some point onward; 43 is the first natural number whose terms don't respect this relation.
%C A228932 The numbers in range 1 .. 200 that exhibit this behavior are 43, 44, 46, 53, 58, 61, 67, 73, 76, 85, 86, 89, 91, 94, 97, 103, 106, 108, 109, 113, 115, 116, 118, 125, 127, 129, 131, 134, 137, 139, 149, 151, 153, 154, 157, 159, 160, 161, 163, 166, 172, 173, 176, 177, 179, 181, 184, 186, 190, 191, 193, 199.
%C A228932 Nevertheless, the expansions of 3*sqrt(43), 9*sqrt(43), and sqrt(43)/5 satisfy the recurrence relation.
%D A228932 See A228931.
%H A228932 G. C. Greubel, <a href="/A228932/b228932.txt">Table of n, a(n) for n = 1..500</a>
%e A228932 sqrt(43) = 6 + 1/2*(1 + 1/9*(1 + 1/30*(1 + 1/60*(1 + 1/122*(1 - 1/878*(1 + ...)))))).
%p A228932 ArticoExp := proc (n, q::posint)::list; local L, i, z; Digits := 50000; L := []; z := frac(evalf(n)); for i to q+1 do if z = 0 then break end if; L := [op(L), round(1/abs(z))*sign(z)]; z := abs(z)*round(1/abs(z))-1 end do; return L end proc
%p A228932 # List the first 8 terms of the expansion of sqrt(43)-6
%p A228932 ArticoExp(sqrt(43),20)
%t A228932 ArticoExp[x_, n_] := Round[1/#] & /@ NestList[Round[1/Abs[#]]*Abs[#] - 1 &, FractionalPart[x], n]; Block[{$MaxExtraPrecision = 50000},
%t A228932 ArticoExp[Sqrt[43] - 6, 20]] (* _G. C. Greubel_, Dec 26 2016 *)
%Y A228932 Cf. A010134, A010497, A228929, A228931.
%K A228932 sign,cofr
%O A228932 1,1
%A A228932 _Giovanni Artico_, Sep 10 2013