OFFSET
0,3
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
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 différentes ..., see p. 8. [Broken link]
Henryk Fukś, 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.
Mathematics Stack Exchange, Is there an integral that proves Pi > 333/106?
Gérard 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.
EXAMPLE
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
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 *)
PROG
(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))<e && !print1(n", ") && e=abs(tan(n))) \\ Illustration of |tan a(n)| -> 0 monotonically. - M. F. Hasler, Apr 01 2013
(Python)
from sympy import pi, continued_fraction_iterator, continued_fraction_convergents
from itertools import islice
cf_iter = continued_fraction_convergents(continued_fraction_iterator(pi))
A002485 = [0, 1] + [f.numerator for f in islice(cf_iter, 25)] # Jwalin Bhatt, Jan 13 2026
CROSSREFS
KEYWORD
nonn,easy,nice,frac
AUTHOR
EXTENSIONS
Extended and corrected by David Sloan, Sep 23 2002
STATUS
approved
