A002945 -id:A002945 - cp's OEIS frontend

A039922 Continued fraction for root of x^5 - x - 1.

1, 5, 1, 42, 1, 3, 24, 2, 2, 1, 16, 1, 11, 1, 1, 2, 31, 1, 12, 5, 1, 7, 11, 1, 4, 1, 4, 2, 2, 3, 4, 2, 1, 1, 11, 1, 41, 12, 1, 8, 1, 1, 1, 1, 1, 9, 2, 1, 5, 4, 1, 25, 4, 6, 11, 1, 4, 1, 6, 1, 1, 1, 2, 2, 2, 4, 11, 1, 4, 1, 3, 2, 8, 1, 3, 3, 6, 21, 11, 2, 1, 1, 10, 2, 1, 3

Offset: 0

N. J. A. Sloane

			1.16730397826141868425604589... = 1 + 1/(5 + 1/(1 + 1/(42 + 1/(1 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134. [Annotated scanned copy]
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A160155 Decimal expansion.

{ allocatemem(932245000); default(realprecision, 21000); x=NULL; p=x^5 - x - 1; rs=polroots(p); r=real(rs[1]); c=contfrac(r); for (n=1, 20001, write("b039922.txt", n-1, " ", c[n])); } \\ Harry J. Smith, May 03 2009

contfrac(polrootsreal(x^5-x-1)[1]) \\ Charles R Greathouse IV, Apr 14 2014

A039923 Continued fraction for 2^(1/3) + sqrt(3).

Original entry on oeis.org

2, 1, 123, 1, 1, 3, 1, 1, 5, 7, 2, 3, 2, 4, 1, 18, 5, 1, 13, 3, 3, 3, 4, 1, 69, 2, 1, 1, 7, 1, 1, 3, 1, 1, 13, 2, 5, 2, 1, 3, 1, 2, 38, 3, 1, 2, 1, 1, 2, 1, 5, 1, 1, 1446, 1, 1, 6, 1, 2, 5, 1, 1, 9, 4, 1, 5, 2, 1, 4, 5, 1, 1, 18, 3, 3, 2, 24, 3, 1, 1, 1, 2, 74, 3, 2, 4, 3, 1

Offset: 0

N. J. A. Sloane

2^(1/3) + sqrt(3) = 2^(1/3) + 3^(1/2). - Harry J. Smith, May 09 2009

			2.9919718574637504582946569... = 2 + 1/(1 + 1/(123 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, May 09 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134. [Annotated scanned copy]
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A160331 = Decimal expansion. - Harry J. Smith, May 09 2009

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(2^(1/3)+3^(1/2)); for (n=1, 20001, write("b039923.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 09 2009

A181495 Positions of the incrementally largest terms in continued fraction for 2^(1/3).

Original entry on oeis.org

1, 2, 4, 10, 12, 32, 36, 572, 1991, 20857, 27432, 28763, 155122, 190271, 288108, 484709, 1395499, 9370521, 12918396, 22646948, 49496125, 73469408, 172128260, 645676547

Offset: 1

John M. Campbell, Oct 23 2010

The corresponding records (or high-water marks) in A002945, the continued fraction for 2^(1/3), are {1, 3, 5, 8, 14, 15, 534, 7451, 12737, 22466, 68346, 148017, 217441, 320408, 533679, 4156269, 4886972, 10253793, ...} - see A268515.

It is not known if this sequence is infinite (i.e., whether the continued fraction expansion is bounded). [Davenport]. - N. J. A. Sloane, Feb 07 2016

H. Davenport, The Higher Arithmetic: An Introduction to the Theory of Numbers, Cambridge, 2008.

Cf. A002945, A268515.

Use Max[ContinuedFraction[2^(1/3), n]] for some positive integer n, e.g. Max[ContinuedFraction[2^(1/3), 288108]].
cf = ContinuedFraction[2^(1/3), 20000000]; mx = 0; k = 1; lst = {}; While[k < 20000000, If[ cf[[k]] > mx, mx = cf[[k]]; AppendTo[lst, k]; Print[{k, cf[[k]]}]]; k++ ]; lst (* Robert G. Wilson v, Oct 24 2010 *)

a(19) from Robert G. Wilson v, Oct 24 2010
a(20)-a(21) from Zak Seidov, Feb 08 2016
a(22)-a(24) from Hans Havermann, Feb 08 2016

A110483 Continued fraction for seventh root of 2.

Original entry on oeis.org

1, 9, 1, 1, 1, 1, 5, 46, 1, 3, 2, 1, 1, 3, 1, 1, 2, 1, 22, 48, 1, 1, 5, 4, 1, 1, 1, 1, 1, 1, 2, 8, 1, 6, 1, 21, 1, 1, 1, 1, 1, 6, 1, 1, 3, 3, 1, 1, 2, 2, 2, 3, 1, 26, 1, 16, 1, 4, 21, 1, 2, 1, 1, 1, 5, 3, 7, 21, 3, 1, 1, 1, 8, 1, 8, 1, 4, 1, 24, 1, 3, 1, 6, 1, 2, 1, 5, 5, 6, 1, 12, 1, 8, 2, 2, 1, 3, 1, 1, 2

Offset: 0

Paul Stoeber (pstoeber(AT)uni-potsdam.de), Sep 09 2005

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A002945, A002950.

import Ratio
floorRoot :: Integer -> Integer -> Integer
floorRoot k n | k>=1 && n>=1 = h n where h x = let y=((k-1)*x+n`div`x^(k-1))`div`k in if y (Integer,Rational)
intFrac x = let ((a,b),~(q,r)) = ((numerator x,denominator x),divMod a b) in (q,r%b)
cf :: Rational -> Rational -> [Integer]
cf x y = let ((xi,xf),(yi,yf)) = (intFrac x,intFrac y) in if xi==yi then xi : cf (recip xf) (recip yf) else []
y = 2^512 -- increase to get more terms, decrease to get a quick answer
(k,n) = (7,2) -- compute continued fraction for k-th root of n
main = print (let x = floorRoot k (n*y^k) in cf (x%y) ((x+1)%y))

ContinuedFraction[Surd[2,7],100] (* Harvey P. Dale, Aug 11 2017 *)

A159824 Continued fraction for Pi^Pi (cf. A073233).

Original entry on oeis.org

36, 2, 6, 9, 2, 1, 2, 5, 1, 1, 6, 2, 1, 291, 1, 38, 50, 1, 2, 5, 4, 1, 2, 2, 1, 5, 1, 4, 13, 2, 1, 4, 3, 3, 1, 2, 25, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 1, 43, 1, 2, 7, 3, 1, 1, 1, 2, 4, 2, 1, 1, 3, 1, 3, 3, 2, 2, 16, 3, 5, 2, 1, 5, 2, 1, 10, 1, 1, 3, 1, 13, 1, 1, 3, 1, 10, 4, 1, 1, 1, 38, 1, 2, 2, 1, 1, 3

Offset: 0

Harry J. Smith, Apr 30 2009

nonn

			36.4621596072079117709908260... = 36 + 1/(2 + 1/(6 + 1/(9 + 1/(2 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A001076, A001203, A001204, A002945, A010767, A011002, A011003, A073233, A073238, A179613, A179615, A179616, A179617.

ContinuedFraction[Pi^Pi,200] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2010 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^Pi); for (n=1, 20001, write("b159824.txt", n-1, " ", x[n])); }

Edited by N. J. A. Sloane, Jul 22 2010

A076595 First occurrence of n as a term in the continued fraction for the cube root of 2.

Original entry on oeis.org

1, 15, 2, 7, 4, 64, 56, 10, 59, 14, 148, 18, 117, 12, 32, 638, 578, 112, 229, 371, 218, 91, 878, 2209, 139, 108, 182, 975, 484, 314, 859, 1977, 454, 597, 1016, 205, 1425, 136, 315, 4201, 51, 3661, 1009, 143, 2188, 3532, 381, 550, 151, 786, 2815, 1444, 654

Offset: 1

Benoit Cloitre, Oct 20 2002

Cf. A002945, A032523.

With[{cf=ContinuedFraction[Surd[2,3],4500]},Flatten[Table[Position[cf,n,{1},1],{n,60}]]] (* Harvey P. Dale, Jul 01 2015 *)

default(realprecision, 1500); v=contfrac(2^(1/3)); a(n)=if(n<0,0,s=1; while(abs(n-component(v,s))>0,s++); s)

cp's OEIS Frontend

A039922 Continued fraction for root of x^5 - x - 1.

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A039923 Continued fraction for 2^(1/3) + sqrt(3).

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A181495 Positions of the incrementally largest terms in continued fraction for 2^(1/3).

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A110483 Continued fraction for seventh root of 2.

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Haskell

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A159824 Continued fraction for Pi^Pi (cf. A073233).

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Mathematica

PARI

Extensions

A076595 First occurrence of n as a term in the continued fraction for the cube root of 2.

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Mathematica

PARI