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.

A019683 Decimal expansion of Pi/16.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Examples

			Pi/16 = 0.19634954084936207740391521145496893026232308746094411381... - _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4.2, p. 494.

Links

Crossrefs

Cf. A157332, A000796, A003881, A019669, A019670, A019675, A019679, A244978.

Programs

Formula

From Peter Bala, Oct 27 2019: (Start)
Equals Integral_{x = 0..1} x^2*sqrt(1 - x^2) dx = Integral_{x = 0..1} x^3*sqrt(1 - x^8) dx.
Equals Integral_{x = 0..inf} x^2/(1 + x^2)^3 dx. (End)
From Amiram Eldar, Aug 04 2020: (Start)
Equals Sum_{k>=1} sin(k)^3 * cos(k)/k.
Equals Sum_{k>=1} sin(k)^3 * cos(k)^2/k.
Equals Sum_{k>=1} (-1)^(k+1) * sin((2*k-1)/4)/(2*k-1)^2. (End)

A072172 a(n) = (2*n+1)*5^(2*n+1).

Original entry on oeis.org

5, 375, 15625, 546875, 17578125, 537109375, 15869140625, 457763671875, 12969970703125, 362396240234375, 10013580322265625, 274181365966796875, 7450580596923828125, 201165676116943359375, 5401670932769775390625, 144354999065399169921875
Offset: 0

Views

Author

N. J. A. Sloane, Jun 30 2002

Keywords

Comments

J. Machin (died 1751) used Pi/4 = 4*Sum_{n=0..inf} (-1)^n/((2*n+1)*5^(2*n+1)) - Sum_{n=0..inf} (-1)^n/((2*n+1)*239^(2*n+1)) to calculate Pi to 100 decimal places.

References

  • H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 73

Links

Crossrefs

Cf. A072173.
Cf. A157332. - Jaume Oliver Lafont, Mar 03 2009

Programs

  • GAP
    List([0..20], n-> (2*n+1)*5^(2*n+1)); # G. C. Greubel, Aug 26 2019
  • Magma
    [(2*n+1)*5^(2*n+1): n in [0..20]]; // G. C. Greubel, Aug 26 2019
  • Maple
    seq((2*n+1)*5^(2*n+1), n=0..20); # G. C. Greubel, Aug 26 2019
  • Mathematica
    Table[(2*n+1)*5^(2*n+1), {n,0,20}] (* G. C. Greubel, Aug 26 2019 *)
  • PARI
    Vec(5*(1+25*x)/(1-25*x)^2 + O(x^20)) \\ Colin Barker, Aug 25 2016
  • PARI
    vector(20, n, (2*n-1)*5^(2*n-1) ) \\ G. C. Greubel, Aug 26 2019
  • Sage
    [(2*n+1)*5^(2*n+1) for n in (0..20)] # G. C. Greubel, Aug 26 2019

Formula

From Colin Barker, Aug 25 2016: (Start)
a(n) = 50*a(n-1) - 625*a(n-2) for n>1.
G.f.: 5*(1+25*x)/(1-25*x)^2.
(End)
From Ilya Gutkovskiy, Aug 25 2016: (Start)
E.g.f.: 5*(1 + 50*x)*exp(25*x).
Sum_{n>=0} 1/a(n) = arctanh(1/5) = 0.2027325540540821...
Sum_{n>=0} (-1)^n/a(n) = arctan(1/5) = A105532 (End)

A072173 a(n) = (2*n+1)*239^(2*n+1).

Original entry on oeis.org

239, 40955757, 3899056325995, 311805194956024553, 22899374409963958061031, 1598709646931895970271741029, 107923510786468980575690686466147, 7113114068808339968612339655730133985, 460482613887654678993386180604955781138143, 29397724727626925615108413436728112018437968221
Offset: 0

Views

Author

N. J. A. Sloane, Jun 30 2002

Keywords

Comments

J. Machin (died 1751) used Pi/4 = 4*Sum_{n=0..oo} (-1)^n/((2*n+1)*5^(2*n+1)) - Sum_{n=0..oo} (-1)^n/((2*n+1)*239^(2*n+1)) to calculate Pi to 100 decimal places.

References

  • H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 73

Links

Crossrefs

Cf. A072172.
Cf. A157332. - Jaume Oliver Lafont, Mar 03 2009

Programs

  • GAP
    List([0..10], n-> (2*n+1)*239^(2*n+1)); # G. C. Greubel, Aug 26 2019
  • Magma
    [(2*n+1)*239^(2*n+1): n in [0..10]]; // G. C. Greubel, Aug 26 2019
  • Maple
    seq((2*n+1)*239^(2*n+1), n = 0..10); # G. C. Greubel, Aug 26 2019
  • Mathematica
    Table[(2*n+1)*239^(2*n+1), {n,0,10}] (* G. C. Greubel, Aug 26 2019 *)
  • PARI
    Vec(239*(1+57121*x)/(1-57121*x)^2 + O(x^10)) \\ Colin Barker, Aug 25 2016
  • PARI
    vector(10, n, (2*n-1)*239^(2*n-1)) \\ G. C. Greubel, Aug 26 2019
  • Sage
    [(2*n+1)*239^(2*n+1) for n in (0..10)] # G. C. Greubel, Aug 26 2019

Formula

From Colin Barker, Aug 25 2016: (Start)
a(n) = 114242*a(n-1) - 3262808641*a(n-2) for n>1.
G.f.: 239*(1+57121*x) / (1-57121*x)^2. (End)
E.g.f.: m*(1+2*m^2*x)*exp(m^2*x), where m=239. - G. C. Greubel, Aug 26 2019
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.