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.

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)

A157332 Denominators of Egyptian fraction for Pi/16 based on Machin's formula.

Original entry on oeis.org

5, -956, -375, 163823028, 15625, -15596225303980, -546875, 1247220779824098212, 17578125, -91597497639855832244124, -537109375, 6394838587727583881086964116, 15869140625, -431694043145875922302762745864588, -457763671875
Offset: 0

Views

Author

Jaume Oliver Lafont, Feb 27 2009

Keywords

Comments

Machin's formula: Pi/4 = 4*atan(1/5) - atan(1/239).
Sum_{n>=0} 1/a(n) = Pi/16 = atan(1/5) - (1/4)*atan(1/239).

Links

Crossrefs

Cf. A019683, A072172, A072173.

Programs

  • GAP
    a:= function(n)
    if n mod 2=0 then return (-1)^(n/2)*(n+1)*5^(n+1);
    else return -4*(-1)^((n-1)/2)*n*(239)^n;
    fi;
  end;
List([0..15], n-> a(n) ); # G. C. Greubel, Aug 26 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 15); Coefficients(R!( 5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2 )); // G. C. Greubel, Aug 26 2019
  • Maple
    seq(coeff(series(5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2, x, n+1), x, n), n = 0..15); # G. C. Greubel, Aug 26 2019
  • Mathematica
    CoefficientList[Series[5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2, {x,0,15}], x] (* G. C. Greubel, Aug 26 2019 *)
  • PARI
    my(x='x+O('x^15)); Vec(5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2) \\ G. C. Greubel, Aug 26 2019
  • Sage
    def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2 ).list()
A077952_list(15) # G. C. Greubel, Aug 26 2019

Formula

a(2n) = (2*n+1)*5^(2*n+1)*(-1)^n,
a(2n+1) = -4*(2*n+1)*239^(2*n+1)*(-1)^n.
G.f.: 5*(1-25*x^2)/(1+25*x^2)^2 - 956*x*(1-57121*x^2)/(1+57121*x^2)^2

Extensions

More terms from Colin Barker, Aug 07 2013
Showing 1-2 of 2 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.