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 1-2 of 2 results.

A101464 Decimal expansion of sqrt(2-sqrt(2)), edge length of a regular octagon with circumradius 1.

Original entry on oeis.org

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

Views

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jan 20 2005

Keywords

Examples

			0.765366864730179543456919968060797733522689124971254082867601271255092067920...

Links

Crossrefs

Cf. A047621, A101465, A179260 (sqrt(2+sqrt(2))), A182168, A285871, A329246.

Programs

Formula

Equals i^(3/4) + i^(-3/4). - Gary W. Adamson, Jul 07 2022
Equals 2*sin(Pi/8) = 2*A182168. - Amiram Eldar, Apr 06 2023
Equals Product_ {k >= 0} ((8*k - 2)*(8*k + 10))/((8*k - 5)*(8*k + 13)). - Antonio Graciá Llorente, Mar 11 2024
Equals Product_{k>=1} (1 + (-1)^k/A047621(k)). - Amiram Eldar, Nov 22 2024
Equals sqrt(A101465) = 1/A285871 = exp(-A329246). - Hugo Pfoertner, Nov 22 2024

A329247 Decimal expansion of Sum_{k>=1} cos(k*Pi/6)/k.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 09 2019

Keywords

Comments

Sum_{k>=1} cos(k*x)/k = Re(Sum_{k>=1} exp(k*x*i)/k) = Re(-log(1-exp(x*i))) = -log(2*|sin(x/2)|), x != 2*m*Pi, where i is the imaginary unit.
In general, for real s and complex z, let f(s,z) = Sum_{k>=1} z^k/k^s, then:
(a) if s <= 0, then f(s,z) converges to Polylog(s,z) if |z| < 1;
(b) if 0 < s <= 1, then f(s,z) converges to Polylog(s,z) if z != 1;
(c) if s > 1, then f(s,z) converges to Polylog(s,z) if |z| <= 1.
As a result, let z = e^(i*x), then the series Sum_{k>=1} (cos(k*x) + i*sin(k*x))/k^s converges to Polylog(s,e^(i*x)) if and only if s > 1, or 0 < s <= 1 and x != 2*m*Pi.

Examples

			0.65847894846240835431252317365398422201349098573375...

Links

Crossrefs

Similar sequences:
A263192 (Sum_{k>=1} cos(k)/sqrt(k) = Re(Polylog(1/2,exp(i))));
A263193 (Sum_{k>=1} sin(k)/sqrt(k) = Im(Polylog(1/2,exp(i))));
this sequence (Sum_{k>=1} cos(k*Pi/6)/k = Re(Polylog(1,exp(i*Pi/6))));
A121225 (Sum_{k>=1} cos(k)/k = Re(Polylog(1,exp(i))));
A329246 (Sum_{k>=1} cos(k*Pi/4)/k = Re(Polylog(1,exp(i*Pi/4))));
A096444 (Sum_{k>=1} sin(k)/k = Im(Polylog(1,exp(i))));
A122143 (Sum_{k>=1} cos(k)/k^2 = Re(Polylog(2,exp(i))));
A096418 (Sum_{k>=1} sin(k)/k^2 = Im(Polylog(2,exp(i)))).

Programs

  • Maple
    Digits := 100: (log(2 + sqrt(3))/2)*10^91:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Nov 09 2019
  • Mathematica
    RealDigits[Log[2 + Sqrt[3]]/2, 10, 100][[1]] (* Amiram Eldar, Dec 05 2021 *)
  • PARI
    default(realprecision, 100); log(2 + sqrt(3))/2

Formula

Equals log(2 + sqrt(3))/2.
Equals -log(2*sin(Pi/12)).
Equals arccoth(sqrt(3)). - Amiram Eldar, Dec 05 2021
From Amiram Eldar, Mar 26 2022: (Start)
Equals arcsinh(1/sqrt(2)).
Equals Sum_{n>=1} arcsinh(1/(sqrt(2^(n+2)+2)+sqrt(2^(n+1)+2))) (Vălean, 2106). (End)
log(2 + sqrt(3))/2 = Sum_{n >= 1} 1/(n*P(n, sqrt(3))*P(n-1, sqrt(3))), where P(n, x) denotes the n-th Legendre polynomial. The first ten terms of the series gives the approximation log(2 + sqrt(3))/2 = 0.658478948(35...) correct to 9 decimal places. - Peter Bala, Mar 16 2024
Showing 1-2 of 2 results.

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