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A228934 Optimal ascending continued fraction expansion of sqrt(44) - 6. 2
2, 4, 15, -99, -199, -800, -79201, -316808, -12545596801, -50182387208, -314783998186522867201, -1259135992746091468808, -198177931028585663493396958369763763148801, -792711724114342653973587833479055052595208 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

See A228929 for the definition of "optimal ascending continued fraction".

This is the first number whose expansion exhibits (in the first 20 terms) a different recurrence relation from that described in A228931.

Conjecture: The terms of the expansion of sqrt(x) are all negative starting from a(4) and satisfy these recurrence relations for n>=3: a(2n) = 4*a(2n-1) - 4 and a(2n+1) = -2*a(2n-1)^2 + 1.

Numbers (in the range 1..1000) that exhibit this recurrence starting from some n are 44, 125, 154, 160, 176, 207, 208, 280, 352, 384, 459, 468, 500, 608, 616, 640, 665, 686, 704, 768, 800, 832, 864, 874, 875, 924.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..21

FORMULA

a(2n) = 4*a(2n-1) - 4 and a(2n+1) = -2*a(2n-1)^2 + 1 for n >= 3.

EXAMPLE

sqrt(44) = 6 + 1/2*(1 + 1/4*(1 + 1/15*(1 - 1/99*(1 - 1/199*(1 - 1/800*(1 - 1/79201*(1 - 1/316808*(1 - 1/12545596801*(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 20 terms of the expansion of sqrt(44)-6

ArticoExp(sqrt(44), 20)

MATHEMATICA

ArticoExp[x_, n_] := Round[1/#] & /@ NestList[Round[1/Abs[#]]*Abs[#] - 1 &, FractionalPart[x], n]; Block[{$MaxExtraPrecision = 50000}, ArticoExp[Sqrt[44] - 6, 20]] (* G. C. Greubel, Dec 26 2016 *)

CROSSREFS

Cf. A228929, A228931, A228932.

Sequence in context: A140836 A020134 A307085 * A120490 A003514 A065598

Adjacent sequences:  A228931 A228932 A228933 * A228935 A228936 A228937

KEYWORD

sign,cofr

AUTHOR

Giovanni Artico, Sep 11 2013

EXTENSIONS

Minor typos corrected by Giovanni Artico, Sep 24 2013

STATUS

approved

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Last modified January 19 03:54 EST 2020. Contains 331031 sequences. (Running on oeis4.)