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A002485 Numerators of convergents to Pi.
(Formerly M3097 N1255)
26
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

Contribution 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

REFERENCES

P. Beckmann, A History of Pi. Golem Press, Boulder, CO, 2nd ed., 1971, p. 171 (but beware errors).

E. B. Burger, Diophantine Olympics ..., Amer. Math. Monthly, 107 (Nov. 2000), 822-829.

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

T. D. Noe, Table of n, a(n) for n=0..201

Marc Daumas, Des implantations differentes ..., see p. 8.

Henryk Fuks, Adam Adamandy Kochanski's approximations of pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739, 2011. 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) for(i=1, #cf=contfrac(Pi), print1(contfracpnqn(vecextract(cf, 2^i-1))[1, 1]", ")) \\ - M. F. Hasler, Apr 01 2013

(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.

Sequence in context: A102223 A189897 A046947 * A193193 A099750 A219268

Adjacent sequences:  A002482 A002483 A002484 * A002486 A002487 A002488

KEYWORD

nonn,easy,nice,frac

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Extended and corrected by David Sloan, Sep 23, 2002.

STATUS

approved

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Last modified November 21 11:17 EST 2014. Contains 249777 sequences.