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 1-7 of 7 results.

A002950 Continued fraction for fifth root of 2.

Original entry on oeis.org

1, 6, 1, 2, 1, 1, 1, 3, 25, 1, 4, 3, 3, 7, 52, 1, 2, 3, 2, 15, 2, 2, 4, 16, 2, 7, 1, 1, 1, 10, 21, 1, 1, 1, 141, 2, 4, 1, 4, 2, 1, 1, 17, 1, 3, 3, 4, 1, 3, 1, 3, 2, 1, 1, 2, 33, 1, 6, 6, 1, 2, 4, 1, 21, 1, 3, 3, 8, 10, 1, 46, 6, 1, 10, 1, 1, 1, 1, 2, 11, 1, 3, 1
Offset: 0

Views

Author

N. J. A. Sloane, Herman P. Robinson

Keywords

Examples

			2^(1/5) = 1.148698354997035006798626946... = 1 + 1/(6 + 1/(1 + 1/(2 + 1/(1 + ...)))). - _Harry J. Smith_, May 12 2009

References

  • H. P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005531 (decimal expansion).
Cf. A002361, A002362 (convergents).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); ContinuedFraction(2^(1/5)); // G. C. Greubel, Nov 02 2018
  • Maple
    with(numtheory):
cfrac(2^(1/5),100,'quotients'); # Muniru A Asiru, Nov 02 2018
  • Mathematica
    ContinuedFraction[2^(1/5), 100] (* G. C. Greubel, Nov 02 2018 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 21000); x=contfrac(2^(1/5)); for (n=1, 20000, write("b002950.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 12 2009

Extensions

Offset changed by Andrew Howroyd, Jul 05 2024

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

A184909 a(n) = n + floor(n*s/r) + floor(n*t/r), where r=2^(1/5), s=r^2, t=r^3.

Original entry on oeis.org

3, 6, 9, 13, 16, 19, 24, 27, 30, 34, 37, 40, 44, 48, 51, 55, 58, 61, 65, 68, 72, 76, 79, 82, 85, 89, 93, 96, 100, 103, 106, 110, 113, 117, 121, 124, 127, 131, 134, 137, 142, 145, 148, 152, 155, 158, 162, 166, 169, 172, 176, 179, 182, 187, 190, 193, 197, 200, 203, 207
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2011

Keywords

Crossrefs

Cf. A005531, A005533, A011093.
Cf. A184910, A184911, A378142, A378185, A379510.

Programs

  • Mathematica
    r = 2^(1/5); s = r^2; t = r^3;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 120}]  (* A184909 *)
Table[b[n], {n, 1, 120}]  (* A184910 *)
Table[c[n], {n, 1, 120}]  (* A184911 *)
(* Clark Kimberling, Jan 18 2025 *)

Extensions

Definition in name corrected by Clark Kimberling, Jan 18 2025

A011183 Decimal expansion of 5th root of 98.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Links

Programs

  • Mathematica
    RealDigits[Surd[98,5],10,120][[1]] (* Harvey P. Dale, Mar 06 2014 *)

Formula

Equals A005531*A011092^2 . - R. J. Mathar, Aug 27 2024

A247584 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 3*a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 13, 43, 113, 253, 509, 969, 1849, 3719, 8009, 18027, 40897, 91257, 198697, 423777, 894081, 1886011, 4007301, 8594411, 18560081, 40181493, 86872293, 187197193, 402060793, 861827743, 1846685729, 3960390059, 8504658049, 18283290609, 39325827729
Offset: 0

Views

Author

Alexander Samokrutov, Sep 20 2014

Keywords

Comments

a(n)/a(n-1) tends to 2.1486... = 1 + 2^(1/5), the real root of the polynomial x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 3.
If x^5 = 2 and n >= 0, then there are unique integers a, b, c, d, g such that (1 + x)^n = a + b*x + c*x^2 + d*x^3 + g*x^4. The coefficient a is a(n) (from A052102). - Alexander Samokrutov, Jul 11 2015
If x=a(n), y=a(n+1), z=a(n+2), s=a(n+3), t=a(n+4) then x, y, z, s, t satisfies Diophantine equation (see link). - Alexander Samokrutov, Jul 11 2015

Links

Crossrefs

The following sequences belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A052101, A052102, A052103, A063727, A083098, A083099, A083100, A084057, A093406, A247344.
Cf. A005531.

Programs

  • Magma
    [n le 5 select 1 else 5*Self(n-1) -10*Self(n-2) +10*Self(n-3) -5*Self(n-4) +3*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Jul 11 2015
  • Maple
    m:=50; S:=series( (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 15 2021
  • Mathematica
    LinearRecurrence[{5,-10,10,-5,3}, {1,1,1,1,1}, 50] (* Vincenzo Librandi, Jul 11 2015 *)
  • Maxima
    makelist(sum(2^k*binomial(n,5*k), k, 0, floor(n/5)), n, 0, 50); /* Alexander Samokrutov, Jul 11 2015 */
  • PARI
    Vec((1-x)^4/(1-5*x+10*x^2-10*x^3+5*x^4-3*x^5) + O(x^100)) \\ Colin Barker, Sep 22 2014
  • Sage
    [sum(2^j*binomial(n, 5*j) for j in (0..n//5)) for n in (0..50)] # G. C. Greubel, Apr 15 2021

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 3*a(n-5).
a(n) = Sum_{k=0...floor(n/5)} (2^k*binomial(n,5*k)). - Alexander Samokrutov, Jul 11 2015
G.f.: (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5). - Colin Barker, Sep 22 2014

A358938 Decimal expansion of the real root of 2*x^5 - 1.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Dec 07 2022

Keywords

Comments

This is the reciprocal of A005531.
The other two complex conjugate pairs of roots are obtained, with the present number r = (1/2)^(1/5) and the golden section phi (A001622), from x1 = r*exp(Pi*i*2/5) = r*(phi - 1 + sqrt(2 + phi)*i)/2 = r*(A001622 - 1 + A188593*i)/2 = 0.2690149185... + 0.8279427859...*i, x2 = r*exp(Pi*i*4/5) = r*(-phi + sqrt(3 - phi)*i)/2 = r*(-A001622 + A182007*i)/2 = -0.7042902001... + 0.5116967824...*i.

Examples

			0.87055056329612413913627001747974609897912542434800304824185956850675...

Crossrefs

Cf. A001622, A182007, A188593, A005531, A011101.

Programs

  • Mathematica
    RealDigits[Surd[1/2, 5], 10, 120][[1]] (* Amiram Eldar, Dec 07 2022 *)

Formula

r = (1/2)^(1/5) = 1/A005531.
Equals A011101/2. - Hugo Pfoertner, Mar 24 2025
Showing 1-7 of 7 results.

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

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