cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A005531 Decimal expansion of fifth root of 2.

Original entry on oeis.org

1, 1, 4, 8, 6, 9, 8, 3, 5, 4, 9, 9, 7, 0, 3, 5, 0, 0, 6, 7, 9, 8, 6, 2, 6, 9, 4, 6, 7, 7, 7, 9, 2, 7, 5, 8, 9, 4, 4, 3, 8, 5, 0, 8, 8, 9, 0, 9, 7, 7, 9, 7, 5, 0, 5, 5, 1, 3, 7, 1, 1, 1, 1, 8, 4, 9, 3, 6, 0, 3, 2, 0, 6, 2, 5, 3, 5, 1, 3, 0, 5, 6, 8, 1, 1, 4, 7, 3, 1, 1, 3, 0, 1, 1, 5, 0, 8, 4, 7, 3, 9, 1, 4, 5, 7
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The sine of 2017 times this number is the near-integer 0.999999999999999978567771261.... - Alonso del Arte, May 03 2013
With the present number r = 2^(1/5) and the golden section phi = A001622 the other (complex) roots of x^5 - 2 are given by x1 = r*exp(2*Pi*i/5) = r*(phi - 1 + sqrt(2 + phi)*i)/2 = r*(A001622 - 1 + A188593*i)/2 = 0.3549673131... + 1.0924770557...*i, x2 = r*exp(4*Pi*i/5) = r*(-phi + sqrt(3 - phi)*i)/2 = r*(-A001622 + A182007*i)/2 = -0.9293164906... + 0.6751879523...*i, and their complex conjugates. - Wolfdieter Lang, Dec 06 2022

Examples

			1.148698354997035006798626946777927589443850889097797505513711118493603....

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002950 (continued fraction).
Cf. A002580 (cube root of 2).
Cf. A001622, A182007, A188593.

Programs

  • Mathematica
    RealDigits[N[2^(1/5),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2012 *)
RealDigits[Surd[2,5],10,120][[1]] (* Harvey P. Dale, May 08 2025 *)
  • PARI
    { default(realprecision, 20080); x=2^(1/5); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b005531.txt", n, " ", d)); } \\ Harry J. Smith, May 12 2009

Formula

Equals Product_{k>=0} (1 + (-1)^k/(5*k + 4)). - Amiram Eldar, Jul 25 2020
From Peter Bala, Mar 02 2022: (Start)
Equals (3/2)*Sum_{n >= 0} (1/(5*n+2) - 1/(5*n-3))*binomial(1/5,n). Cf. A002580.
Equals (5/4)*hypergeom([-1/5, -3/5], [7/5], -1). (End)

Extensions

More terms from Olaf Voß, Feb 13 2008

A002361 Denominators of continued fraction convergents to fifth root of 2.

Original entry on oeis.org

1, 6, 7, 20, 27, 47, 74, 269, 6799, 7068, 35071, 112281, 371914, 2715679, 141587222, 144302901, 430193024, 1434881973, 3299956970, 50934236523, 105168430016, 261271096555, 1150252816236, 18665316156331, 38480885128898, 288031512058617, 326512397187515
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
  • P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002362 (numerators), A002950, A005531.

Programs

  • Mathematica
    Denominator[Convergents[2^(1/5), 30]] (* Harvey P. Dale, Apr 27 2012 *)
  • PARI
    a(n)= contfracpnqn(contfrac(2^(1/5), n+1))[2, 1]; \\ Michel Marcus, Sep 07 2013

Extensions

More terms from Herman P. Robinson
Definition clarified and more terms from Harvey P. Dale, Apr 27 2012
Offset changed by Andrew Howroyd, Jul 05 2024

A002362 Numerators of continued fraction convergents to fifth root of 2.

Original entry on oeis.org

1, 7, 8, 23, 31, 54, 85, 309, 7810, 8119, 40286, 128977, 427217, 3119496, 162641009, 165760505, 494162019, 1648246562, 3790655143, 58508073707, 120806802557, 300121678821, 1321293517841, 21440817964277, 44202929446395, 330861324089042, 375064253535437
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
  • P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002361 (denominators), A002950, A005531.

Programs

  • Mathematica
    Numerator[Convergents[2^(1/5), 30]] (* Vincenzo Librandi, Sep 08 2013 *)
  • PARI
    a(n)= contfracpnqn(contfrac(2^(1/5), n+1))[1, 1]; \\ Michel Marcus, Sep 07 2013

Extensions

More terms from Herman P. Robinson
More terms from Michel Marcus, Sep 07 2013
a(26)-a(27) corrected by Vincenzo Librandi, Sep 08 2013
Offset changed by Andrew Howroyd, Jul 05 2024

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

Views

Author

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

Keywords

Links

Crossrefs

Cf. A002945, A002950.

Programs

  • Haskell
    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))
  • Mathematica
    ContinuedFraction[Surd[2,7],100] (* Harvey P. Dale, Aug 11 2017 *)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.