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.

A019425 Continued fraction for tan(1/2).

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, 24, 1, 28, 1, 32, 1, 36, 1, 40, 1, 44, 1, 48, 1, 52, 1, 56, 1, 60, 1, 64, 1, 68, 1, 72, 1, 76, 1, 80, 1, 84, 1, 88, 1, 92, 1, 96, 1, 100, 1, 104, 1, 108, 1, 112, 1, 116, 1, 120, 1, 124, 1, 128, 1, 132, 1, 136, 1, 140, 1, 144, 1, 148, 1, 152, 1, 156, 1
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

From Peter Bala, Nov 17 2019: (Start)
The simple continued fraction expansion for tan(1/2) may be derived by setting z = 1/2 in Lambert's continued fraction tan(z) = z/(1 - z^2/(3 - z^2/(5 - ... ))) and, after using an equivalence transformation, making repeated use of the identity 1/(n - 1/m) = 1/((n - 1) + 1/(1 + 1/(m - 1))).
The same approach produces the simple continued fraction expansions for the numbers tan(1/n), n*tan(1/n) and 1/n*tan(1/n) for n = 1,2,3,.... [added Oct 03 2023: and, even more generally, the simple continued fraction expansions for the numbers d*tan(1/n) and 1/d*tan(1/n), where d divides n. See A019429 for an example]. (End)

Examples

			0.546302489843790513255179465... = 0 + 1/(1 + 1/(1 + 1/(4 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 13 2009

Links

Crossrefs

Cf. A161011 (decimal expansion). Cf. A019426 through A019433.

Programs

  • Magma
    [0,1] cat [n-1/2-(n-3/2)*(-1)^n+Binomial(1,n)- 2*Binomial(0,n): n in [2..80]]; // Vincenzo Librandi, Jan 03 2016
  • Maple
    a := n -> if n < 2 then n else ifelse(irem(n, 2) = 0, 1, 2*n - 2) fi:
seq(a(n), n = 0..80);  # Peter Luschny, Oct 03 2023
  • Mathematica
    Join[{0, 1}, LinearRecurrence[{0, 2, 0, -1}, {1, 4, 1, 8}, 100]] (* Vincenzo Librandi, Jan 03 2016 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 85000); x=contfrac(tan(1/2)); for (n=0, 20000, write("b019425.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 13 2009

Formula

a(n) = n - 1/2 - (n-3/2)*(-1)^n + binomial(1,n) - 2*binomial(0,n). - Paul Barry, Oct 25 2007
From Philippe Deléham, Feb 10 2009: (Start)
a(n) = 2*a(n-2) - a(n-4), n>=6.
G.f.: (x + x^2 + 2*x^3 - x^4 + x^5)/(1-x^2)^2. (End)
From Peter Bala, Nov 17 2019; (Start)
Related simple continued fraction expansions:
2*tan(1/2) = [1, 10, 1, 3, 1, 26, 1, 7, 1, 42, 1, 11, 1, 58, 1, 15, 1, 74, 1, 19, 1, 90, ...]
(1/2)*tan(1/2) = [0; 3, 1, 1, 1, 18, 1, 5, 1, 34, 1, 9, 1, 50, 1, 13, 1, 66, 1, 17, 1, 82, ...]. (End)

A019426 Continued fraction for tan(1/3).

Original entry on oeis.org

0, 2, 1, 7, 1, 13, 1, 19, 1, 25, 1, 31, 1, 37, 1, 43, 1, 49, 1, 55, 1, 61, 1, 67, 1, 73, 1, 79, 1, 85, 1, 91, 1, 97, 1, 103, 1, 109, 1, 115, 1, 121, 1, 127, 1, 133, 1, 139, 1, 145, 1, 151, 1, 157, 1, 163, 1, 169, 1, 175, 1, 181, 1, 187, 1, 193, 1, 199, 1, 205, 1, 211, 1, 217, 1, 223, 1
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

The simple continued fraction expansion of 3*tan(1/3) is [1; 25, 1, 3, 1, 61, 1, 7, 1, 97, 1, 11, 1, ..., 36*n + 25, 1, 4*n + 3, 1, ...], while the simple continued fraction expansion of (1/3)*tan(1/3) is [0; 8, 1, 1, 1, 43, 1, 5, 1, 79, 1, 9, 1, 115, 1, 13, 1, ..., 36*n + 7, 1, 4*n + 1, 1, ...]. See my comment in A019425. - Peter Bala, Sep 30 2023

Examples

			0.346253549510575491038543565... = 0 + 1/(2 + 1/(1 + 1/(7 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 13 2009

Links

Crossrefs

Cf. A161012 (decimal expansion of tan(1/3)).
Cf. continued fractions for tan(1/m): A019425 (m=2), A019427 (m=4), A019428 (m=5), A019429 (m=6), A019430 (m=7), A019431 (m=8), A019432 (m=9), A019433 (m=10), A093178 (m=1).

Programs

  • Magma
    [n le 1 select 2*n else 1+3*(1-(-1)^n)*(n-1)/2: n in [0..80]]; // Bruno Berselli, Sep 21 2012
  • Mathematica
    ContinuedFraction[Tan[1/3], 80] (* Bruno Berselli, Sep 21 2012 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 88000); x=contfrac(tan(1/3)); for (n=0, 20000, write("b019426.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 13 2009

Formula

From Bruno Berselli, Sep 21 2012: (Start)
G.f.: x*(2+x+3*x^2-x^3+x^4)/(1-x^2)^2.
a(n) = 2*a(n-2)-a(n-4) with n>4, a(0)=0, a(1)=2, a(2)=1, a(3)=7, a(4)=1.
a(n) = 1+3*(1-(-1)^n)*(n-1)/2 with n>1, a(0)=0, a(1)=2.
For k>0: a(2k) = 1, a(4k+1) = 2*a(2k+1)-1 and a(4k+3) = 2*a(2k+1)+5, with a(0)=0, a(1)=2. (End)

A161015 Decimal expansion of tan(1/6).

Original entry on oeis.org

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

Views

Author

Harry J. Smith, Jun 13 2009

Keywords

Comments

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

Examples

			0.168227218302242461257216080569916100969393651685056685724032739248247...

Links

Crossrefs

Cf. A019429 Continued fraction.

Programs

  • Mathematica
    RealDigits[Tan[1/6],10,120][[1]] (* Harvey P. Dale, Aug 26 2014 *)
  • PARI
    default(realprecision, 20080); x=10*tan(1/6); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b161015.txt", n, " ", d));
Showing 1-3 of 3 results.

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