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A002485 Numerators of convergents to Pi.
(Formerly M3097 N1255)
37
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 (list; graph; refs; listen; history; text; internal format)
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' experimentally obtained identities:

  (-1)^n*(Pi-A002485(n)/A002486(n)) = 1/abs(i)*2^j)*Integrate(x^l*(1-x)^m*(k+(k+i)*x^2)/(1+x^2),x= 0..1) where {i, j, k, l, m} are some integers (see the StackExchange 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).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..201 from T. D. Noe)

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

G. P. Michon, Continued Fractions

StackExchange, Is there an integral that proves pi > 333/106

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

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

CROSSREFS

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

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

Sequence in context: A189897 A306578 A046947 * A193193 A099750 A219268

Adjacent sequences:  A002482 A002483 A002484 * A002486 A002487 A002488

KEYWORD

nonn,easy,nice,frac,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Extended and corrected by David Sloan, Sep 23 2002

STATUS

approved

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Last modified October 20 05:26 EDT 2020. Contains 337897 sequences. (Running on oeis4.)