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

2, 9, 30, 60, 122, -878, 11429, 35241, -177141, 709582, -3123032, -1157723745, 3237738813, -16178936725, 33395053634, -71863018424, -153349368674, -386763022623, -8021033029400, 16314606875900, 52522689388692

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 A352405 A056778

Adjacent sequences: A228929 A228930 A228931 * A228933 A228934 A228935

Keyword

sign,cofr

Author

Giovanni Artico, Sep 10 2013

Status

approved