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 A228932 Optimal ascending continued fraction expansion of sqrt(43) - 6. 4
 2, 9, 30, 60, 122, -878, 11429, 35241, -177141, 709582, -3123032, -1157723745, 3237738813, -16178936725, 33395053634, -71863018424, -153349368674, -386763022623, -8021033029400, 16314606875900, 52522689388692 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS See A228929 for the definition of "optimal ascending continued fraction". 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. 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. Nevertheless, the expansions of 3*sqrt(43), 9*sqrt(43), and sqrt(43)/5 satisfy the recurrence relation. REFERENCES See A228931. LINKS G. C. Greubel, Table of n, a(n) for n = 1..500 EXAMPLE sqrt(43) = 6 + 1/2*(1 + 1/9*(1 + 1/30*(1 + 1/60*(1 + 1/122*(1 - 1/878*(1 + ...)))))). MAPLE 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 # List the first 8 terms of the expansion of sqrt(43)-6 ArticoExp(sqrt(43), 20) MATHEMATICA ArticoExp[x_, n_] := Round[1/#] & /@ NestList[Round[1/Abs[#]]*Abs[#] - 1 &, FractionalPart[x], n]; Block[{\$MaxExtraPrecision = 50000}, ArticoExp[Sqrt[43] - 6, 20]] (* G. C. Greubel, Dec 26 2016 *) CROSSREFS Cf. A010134, A010497, A228929, A228931. Sequence in context: A042357 A079783 A182975 * A196421 A056778 A177111 Adjacent sequences:  A228929 A228930 A228931 * A228933 A228934 A228935 KEYWORD sign,cofr AUTHOR Giovanni Artico, Sep 10 2013 STATUS approved

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Last modified November 27 01:16 EST 2021. Contains 349344 sequences. (Running on oeis4.)