A002485 Numerators of convergents to Pi.

0, 1, 3, 22, 333, 355, 103993, 104348, 208341, 312689, 833719, 1146408, 4272943, 5419351, 80143857, 165707065, 245850922, 411557987, 1068966896, 2549491779, 6167950454, 14885392687, 21053343141, 1783366216531, 3587785776203, 5371151992734, 8958937768937

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• frac

Comments

From Alexander R. Povolotsky, Apr 09 2012: (Start)

K. S. Lucas found, by brute-force search, using Maple programming, several different variants of integral identities which relate each of several first Pi convergents (A002485(n)/A002486(n)) to Pi.

I conjecture the following identity below, which represents a generalization of Stephen Lucas's experimentally obtained identities:

(-1)^n*(Pi-A002485(n)/A002486(n)) = (1/abs(i)*2^j)*Integral_{x=0..1} (x^l*(1-x)^m*(k+(k+i)*x^2)/(1+x^2)) dx where {i, j, k, l, m} are some integers (see the Mathematics Stack Exchange link below). (End)

From a(1)=1 on also: Numbers for which |tan x| decreases monotonically to zero, in the same spirit as A004112, A046947, ... - M. F. Hasler, Apr 01 2013

See also A332095 for n*|tan n| < 1. - M. F. Hasler, Sep 13 2020

Examples

			The convergents are 0, 1, 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317, 312689/99532, 833719/265381, 1146408/364913, 4272943/1360120, 5419351/1725033, 80143857/25510582, 165707065/52746197, 245850922/78256779, 411557987/131002976, 1068966896/340262731, 2549491779/811528438,  ... = A002485/A002486
		

References

• P. Beckmann, A History of Pi. Golem Press, Boulder, CO, 2nd ed., 1971, p. 171 (but beware errors).
• CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
• P. Finsler, Über die Faktorenzerlegung natuerlicher Zahlen, Elemente der Mathematik, 2 (1947), 1-11, see p. 7.
• K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 293.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 274.

Links

• Daniel Mondot, Table of n, a(n) for n = 0..1947 (terms 0..201 from T. D. Noe, terms 202..1000 from G. C. Greubel).
• E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
• Marc Daumas, Des implantations differentes ..., see p. 8. [Broken link]
• Henryk Fuks, Adam Adamandy Kochanski's approximations of Pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739 [math.HO], 2011-2014. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45.
• S. K. Lucas, Integral approximations to Pi with nonnegative integrands
• Mathematics Stack Exchange, Is there an integral that proves pi > 333/106
• G. P. Michon, Continued Fractions
• Eric Weisstein's World of Mathematics, Pi
• Eric Weisstein's World of Mathematics, Pi Continued Fraction
• Eric Weisstein's World of Mathematics, Pi Approximations
• Index entries for sequences related to the number Pi

Crossrefs

Cf. A002486 (denominators), A046947, A072398/A072399.

Cf. A096456 (numerators of convergents to Pi/2).

Programs

Maple

Digits := 60: E := Pi; convert(evalf(E),confrac,50,'cvgts'): cvgts;

Mathematica

Join[{0, 1}, Numerator @ Convergents[Pi,29]] (* Jean-François Alcover, Apr 08 2011 *)

PARI

contfracpnqn(cf=contfrac(Pi),#cf)[1,] \\ M. F. Hasler, Apr 01 2013, simplified Oct 13 2020

PARI

e=9e9;for(n=1,1e9,abs(tan(n)) 0 monotonically. - M. F. Hasler, Apr 01 2013

Extensions

Extended and corrected by David Sloan, Sep 23 2002