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 1-3 of 3 results.

A078976 Numerator of n-th convergent to e^(2/3).

Original entry on oeis.org

1, 2, 37, 261, 298, 559, 5888, 318511, 5102064, 5420575, 10522639, 205350716, 18492087079, 462507527691, 480999614770, 943507142461, 26899199603678, 3390242657205889, 115295149544603904, 118685392201809793, 233980541746413697, 8775965436819116582, 1421940381306443299981
Offset: 1

Views

Author

Benoit Cloitre, Dec 19 2002

Keywords

Crossrefs

Cf. A069951, A001518, A007676, A078977 (denominators).

Programs

  • Mathematica
    Convergents[Exp[2/3], 25] // Numerator (* Amiram Eldar, May 09 2025 *)
  • PARI
    default(realprecision,100); /* large enough */
a(n)=contfracpnqn(contfrac(exp(2/3), n))[1,1]
vector(30,n,a(n))

Formula

Special cases : a(5k+1) = A001518(3k); a(5k+3) = A001518(3k+2).

A078977 Denominator of n-th convergent to e^(2/3).

Original entry on oeis.org

1, 1, 19, 134, 153, 287, 3023, 163529, 2619487, 2783016, 5402503, 105430573, 9494154073, 237459282398, 246953436471, 484412718869, 13810509564803, 1740608617884047, 59194503517622401, 60935112135506448
Offset: 1

Views

Author

Benoit Cloitre, Dec 19 2002

Keywords

Crossrefs

Cf. A069951, A065923, A001518, A007677, A078976.

Programs

  • Mathematica
    Denominator[Convergents[E^(2/3),20]] (* Harvey P. Dale, Dec 01 2013 *)
  • PARI
    a(n)=component(component(contfracpnqn(contfrac(exp(2/3),n)),1),2) \\ (Warning: this will give only a limited number of correct terms, depending on the precision used. - The Editors, Oct 13 2009. See A078976 for better code.)

Formula

Special cases : a(5k+1)=abs(A065923(3k)); a(5k+3)=abs(A065923(3k+2)) where A065923(n)=y(n, -3) where y(n, x)=sum (k=0, n, (n+k)!*(x/2)^k/((n-k)!*k!))

A078690 Continued fraction expansion of e^(2/5).

Original entry on oeis.org

1, 2, 30, 12, 1, 1, 17, 90, 27, 1, 1, 32, 150, 42, 1, 1, 47, 210, 57, 1, 1, 62, 270, 72, 1, 1, 77, 330, 87, 1, 1, 92, 390, 102, 1, 1, 107, 450, 117, 1, 1, 122, 510, 132, 1, 1, 137, 570, 147, 1, 1, 152, 630, 162, 1, 1, 167, 690, 177, 1, 1, 182, 750, 192, 1, 1, 197, 810, 207
Offset: 0

Views

Author

Benoit Cloitre, Dec 17 2002

Keywords

Links

Crossrefs

Cf. A069951.

Programs

  • Mathematica
    Block[{$MaxExtraPrecision=1000},ContinuedFraction[E^(2/5),70]] (* Harvey P. Dale, Sep 04 2011 *)
  • PARI
    contfrac(exp(2/5))

Formula

For k>=0, a(5k+1)=15k+2 a(5k+2)=60k+30 a(5k+3)=15k+12 a(5k)=a(5k+4)=1.
G.f.: -(x^9-3*x^8-30*x^7-13*x^6+x^5-x^4-12*x^3-30*x^2-2*x-1) / ((x-1)^2*(x^4+x^3+x^2+x+1)^2). - Colin Barker, Jun 24 2013
Showing 1-3 of 3 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.