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.

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A129187 Decimal expansion of arcsinh(1/3).

Original entry on oeis.org

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

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Author

N. J. A. Sloane, Jul 27 2008

Keywords

Comments

Archimedes's-like scheme: set p(0) = 1/sqrt(10), q(0) = 1/3; p(n+1) = 2*p(n)*q(n)/(p(n)+q(n)) (arithmetic mean of reciprocals, i.e., 1/p(n+1) = (1/p(n) + 1/q(n))/2), q(n+1) = sqrt(p(n+1)*q(n)) (geometric mean, i.e., log(q(n+1)) = (log(p(n+1)) + log(q(n)))/2), for n >= 0. The error of p(n) and q(n) decreases by a factor of approximately 4 each iteration, i.e., approximately 2 bits are gained by each iteration. Set r(n) = (2*q(n) + p(n))/3, the error decreases by a factor of approximately 16 for each iteration, i.e., approximately 4 bits are gained by each iteration. For a similar scheme see also A244644. - A.H.M. Smeets, Jul 12 2018

Examples

			0.32745015023725844332253525998825812770052452899076745127562...

Links

Crossrefs

Cf. A002390, A129200, A129269, A244644.

Programs

Formula

Equals log((1 + sqrt(10))/3). - Jianing Song, Jul 12 2018
Equals arccoth(sqrt(10)). - Amiram Eldar, Feb 09 2024

A259441 a(n) is the least number of sides of a regular inscribed k-gon whose perimeter yields Pi to within 1/10^n.

Original entry on oeis.org

3, 8, 23, 72, 228, 719, 2274, 7189, 22733, 71887, 227327, 718869, 2273261, 7188681, 22732604, 71886806, 227326039, 718868054, 2273260386, 7188680533, 22732603855, 71886805327, 227326038545, 718868053265, 2273260385449, 7188680532650, 22732603854487, 71886805326500
Offset: 0

Views

Author

Robert G. Wilson v, Jun 27 2015

Keywords

Comments

Since the perimeter equals n*sin(180º/n), increasing n to greater values will yield a more accurate value of Pi.
Lim n -> inf., a(n+1)/a(n) = sqrt(10). This implies that a(n+2) ~ 10*a(n).
Lim n -> inf., a(2n) = 10^n*sqrt(Pi^3/6) and a(2n+1) = 10^n*sqrt(Pi^3/60).
Lim n -> inf., A259442(n)/a(n) = sqrt(2).

Examples

			a(0) = 3 since the perimeter of an inscribed triangle is sqrt(27)/2 which equals approximately 2.598076... and this is within 1.0 of Pi's true value;
a(1) = 8 since the perimeter of an inscribed octagon is 4*sqrt(2 - sqrt(2)) which equals approximately 3.061467... and this is within 0.1 of Pi's true value;
a(2) = 23 since the perimeter of an inscribed 23-gon is approximately 3.131832... and this in within 0.01 of Pi's true value; etc.

References

  • William H. Beyer, Ed., CRC Standard Mathematical Tables, 27th Ed., IV - Geometry, Mensuration Formulas, p. 122, Boca Raton 1984.
  • Daniel Zwillinger, Editor-in-Chief, 31st Ed., CRC Standard Mathematical Tables and Formulae, 4.5.3 Geometry - Regular Polygons, p. 324, Boca Raton, 2003.
  • Jan Gullberg, Mathematics: From the Birth of Numbers, 13.3 Solving Triangles, p. 479, W. W. Norton & Co., NY, 1997.
  • Catherine A. Gorini, Ph.D., The Facts on File Geometry Handbook, Charts & Tables, p. 262, Checkmark Books, NY, 2005.

Links

Crossrefs

Cf. A000796, A244644, A259442.

Programs

  • Mathematica
    f[n_] := Block[{k = Floor[ Sqrt[ 10]*f[n - 1] - 6]}, While[Pi > k*Sin[Pi/k] + 10^-n, k++]; k]; f[-1] = 3; Array[f, 28, 0]

A259442 a(n) is the least number of sides of a regular circumscribed k-gon whose perimeter yields Pi to within 1/10^n.

Original entry on oeis.org

4, 11, 33, 102, 322, 1017, 3215, 10167, 32149, 101664, 321488, 1016633, 3214876, 10166330, 32148757, 101663296, 321487567, 1016632951, 3214875668, 10166329505, 32148756680, 101663295049, 321487566791, 1016632950485, 3214875667907, 10166329504841, 32148756679070, 101663295048410
Offset: 0

Views

Author

Robert G. Wilson v, Jun 27 2015

Keywords

Comments

Since the perimeter equals n*tan(180º/n), increasing n to greater values will yield a more accurate value of Pi.
Lim n -> inf., a(n+1)/a(n) = sqrt(10). This implies that a(n+2) ~ 10*a(n).
Lim n -> inf., a(2n) = 10^n*sqrt(Pi^3/3) and a(2n+1) = 10^n*sqrt(Pi^3/30).
Lim n -> inf., a(n)/A259441(n) = sqrt(2).

Examples

			a(0) # 3 since the perimeter of the circumscribed triangle is sqrt(27) which equals approximately 5.196152... which exceeds Pi by more than 1;
a(0) = 4 since the perimeter of the circumscribed square is 4 and this is within 1 of the true value of Pi;
a(1) = 11 since the perimeter of the circumscribed 11-gon which equals approximately 3.229891... which is within 0.1 of the true value of Pi;
a(2) = 33 since the perimeter of the circumscribed 33-gon which equals approximately 3.151117... which is within 0.01 of the true value of Pi; etc.

References

  • William H. Beyer, Ed., CRC Standard Mathematical Tables, 27th Ed., IV - Geometry, Mensuration Formulas, p. 122, Boca Raton 1984.
  • Daniel Zwillinger, Editor-in-Chief, 31st Ed., CRC Standard Mathematical Tables and Formulae, 4.5.3 Geometry - Regular Polygons, p. 324, Boca Raton, 2003.
  • Jan Gullberg, Mathematics: From the Birth of Numbers, 13.3 Solving Triangles, p. 479, W. W. Norton & Co., NY, 1997.
  • Catherine A. Gorini, Ph.D., The Facts on File Geometry Handbook, Charts & Tables, p. 262, Checkmark Books, NY, 2005.

Links

Crossrefs

Cf. A000796, A244644, A259441.

Programs

  • Mathematica
    f[n_] := Block[{k = Floor[ Sqrt[ 10]*f[n - 1]] - 6}, While[Pi + 10^-n < k*Tan[Pi/k], k++]; k]; f[-1] = 3; Array[f, 28, 0]
Previous Showing 11-13 of 13 results.

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

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