A002486 -id:A002486 - cp's OEIS frontend

A136121 Integer Log of (Numerators convergent to Pi / Denominators convergent to Pi) = A001414(A002485/A002486) = A001414(A002485)-A001414(A002486).

3, 6, -12, -37, 103369, 1298, 3173, 119, 568338, -2377, 4271976, 3671, 2966929, 4679, 3755, 411547076, -7548356, -742070, 131900, 108983, 13988275, 198923609, 54008, 2194971, 689148781190, 1587713977705, 6700494215, 2691971279, 1226979802842339, -297697344982

Offset: 0

Carlos Alves, Dec 16 2007

The integer log of a fraction p/q is A001414(p) - A001414(q).

Cf. A001414, A002485, A002486.

sopfr = Function[x, Plus @@ Map[Times @@ # &, FactorInteger[x]]]; Table[sopfr[FromContinuedFraction[ContinuedFraction[Pi, k]]], {k, 1, 50}]

A360370 Intersection of A002486 and A360367.

Original entry on oeis.org

1, 7, 106, 113, 33102, 99532, 265381, 1360120, 1725033, 25510582, 78256779, 340262731, 811528438, 1963319607, 6701487259, 1142027682075, 2851718461558, 136308121570117, 1952799169684491, 9627687726852338, 21208174623389167, 136876735467187340, 842468587426513207

Offset: 1

Stefano Spezia, Feb 04 2023

Cf. A002486, A360367.

Cf. A000796, A011557, A360366, A360369 (numerators).

imax=34; Intersection[Denominator[Convergents[ContinuedFraction[Pi, imax]]], Table[Denominator[Rationalize[Pi, 10^(-i)]], {i,0,2*imax}]]

A356665 Number of correct decimal digits of the approximation of Pi obtained from the continued fraction convergents A002485(n)/A002486(n).

Original entry on oeis.org

1, 3, 5, 7, 10, 10, 10, 10, 12, 11, 13, 13, 15, 16, 16, 17, 18, 18, 19, 20, 22, 24, 25, 25, 26, 28, 30, 31, 31, 33, 34, 35, 38, 40, 41, 41, 42, 43, 45, 46, 46, 47, 48, 50, 51, 52, 52, 54, 55, 56, 56, 57, 57, 59, 60, 60, 61, 61, 62, 61, 63, 65, 64

Offset: 2

Daniel Mondot, Aug 21 2022

For most terms the number of correct digits is equal to or slightly less than the sum of the number of digits of the numerator and the denominator.

But for some pairs, the number of correct digits exceeds that sum. For example, a(5) = 7 digits is 1 more than length("355") + length("113") = 6.

			For n=5, A002485(5)/A002486(5) = 355/113 = 3.1415929..., 7 correct decimal digits of Pi. So a(5) = 7.

Daniel Mondot, Table of n, a(n) for n = 2..10000

Cf. A002485, A002486, A000796.

A002485 Numerators of convergents to Pi.

Original entry on oeis.org

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

N. J. A. Sloane

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

			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

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.

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

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

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

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

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

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

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

Extended and corrected by David Sloan, Sep 23 2002

A046947 Numbers k such that |sin(k)| (or |tan(k)| or |sec(k)|) decreases monotonically to 0; also |cos(k)| (or |cosec(k)| or |cot(k)|) increases.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also numerators of convergents to Pi (A002486 gives denominators) beginning at 1.

Integer circumferences of circles with a(0)=1 and a(n+1) is the smallest integer circumference with corresponding diameter nearer an integer than is the diameter of the circle with circumference a(n). See PARI program. - Rick L. Shepherd, Oct 06 2007

			|sin(4272943)| = 0.000000549579497810490800503139..., |tan(4272943)| = 0.000000549579497810573797346111..., |sec(4272943)| = 1.00000000000015101881221...
|cos(4272943)| = 0.999999999999848981187793172965367089856..., |cosec(4272943)| = 1819572.97167010734684889..., |cot(4272943)| = 1819572.97166983255709999...

K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 293.
Suggested by a question from Alan Walker (Alan_Walker(AT)sabre.com)

Michael De Vlieger, Table of n, a(n) for n = 0..1946
Carlos Hernan Lopez Zapata and Nikos Mantzakouras, Diophantine Flint-Hills series: convergence, polylogarithms and pi's bound, ResearchGate (2025). See pp. 6, 69.
Eric Weisstein's World of Mathematics, Cosecant
Eric Weisstein's World of Mathematics, Flint Hills Series

Cf. A004112, A049946. See also A002485, which is the same sequence but begins at 0.

Digits := 50; M := 10000; a := [ 1 ]; R := sin(1.); for n from 2 to M do t1 := evalf(sin(n)); if abs(t1)Zerinvary Lajos, Feb 07 2007

z={}; current=1; Do[ If[ Abs[ Sin[ n]] < current, AppendTo[ z, current=Abs[ Sin[ n]]]], {n, 1, 10^7}]; z (* or *)
Join[{1}, Table[ Numerator[ FromContinuedFraction[ ContinuedFraction[Pi, n]]], {n, 1, 23}]] (* Wouter Meeussen *)
Join[{1},Convergents[Pi,30]//Numerator] (* Harvey P. Dale, May 05 2019 *)

/* Program calculates a(n) without using sin or continued fraction functions */ {d=1/Pi; print1("1, "); for(circum=2,500000000, dm=circum/Pi; dmin=min(dm-floor(dm),ceil(dm)-dm); if(dminRick L. Shepherd, Oct 06 2007

More terms from Wouter Meeussen
Further terms from Michel ten Voorde
Edited and extended by Robert G. Wilson v, Jan 28 2003
Typo in examples fixed by Paolo Bonzini, Mar 21 2012

A068028 Decimal expansion of 22/7.

Original entry on oeis.org

3, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4

Offset: 1

Nenad Radakovic, Mar 22 2002

This is an approximation to Pi. It is accurate to 0.04025%.
Consider the recurring part of 22/7 and the sequences R(i) = 2, 1, 4, 2, 3, 0, 2, ... and Q(i) = 1, 4, 2, 8, 5, 7, 1, .... For i > 0, let X(i) = 10*R(i) + Q(i). Then Q(i+1) = floor(X(i)/Y); R(i+1) = X(i) - Y*Q(i+1); here Y=5; X(0)=X=7. Note 1/7 = 7/49 = X/(10*Y-1). Similar comment holds elsewhere. If we consider the sequences R(i) = 3, 2, 3, 5, 5, 1, 4, 0, 6, 4, 6, 3, 4, 3, 1, 1, 5, 2, 6, 0, 2, 0, 3, ... and Q(i) = A021027, we have X=3; Y=7 (attributed to Vedic literature). - K.V.Iyer, Jun 16 2010, Jun 18 2010
The sequence of convergents of the continued fraction of Pi begins [3, 22/7, 333/106, 355/113, 103993/33102, ...]. 22/7 is the second convergent. The summation 240*Sum_{n >= 1} 1/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)(4*n+6)*(4*n+7)) = 22/7 - Pi shows that 22/7 is an over-approximation to Pi. - Peter Bala, Oct 12 2021

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 187, 239.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi and §13.3 Solving Triangles, pp. 90, 479.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 49.

D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163. - _N. J. A. Sloane_, Mar 24 2012
D. P. Dalzell, On 22/7, J. London Math. Soc. 19, 133-134, 1944.
Dale Winham, Facts about Pi.
Index entries for sequences related to the number Pi.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A068079, A068089, A002485, A002486, A046965, A046947.

I:=[3,1,4,2,8]; [n le 5 select I[n] else Self(n-1)-Self(n-3)+Self(n-4): n in [1..100]]; // Vincenzo Librandi, Mar 27 2015

CoefficientList[Series[(3 - 2 x + 3 x^2 + x^3 + 4 x^4) / ((1 - x) (1 + x) (1 - x + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 27 2015 *)
Join[{3},LinearRecurrence[{1, 0, -1, 1},{1, 4, 2, 8},104]] (* Ray Chandler, Aug 26 2015 *)
RealDigits[22/7,10,120][[1]] (* Harvey P. Dale, Oct 04 2021 *)

a(0)=3, a(n) = floor(714285/10^(5-(n mod 6))) mod 10. - Sascha Kurz, Mar 23 2002 [corrected by Jason Yuen, Aug 18 2024]
Equals 100*A021018 - 4 = 3 + A020806. - R. J. Mathar, Sep 30 2008
For n>1 a(n) = A020806(n-2) (note offset=0 in A020806 and offset=1 in A068028). - Zak Seidov, Mar 26 2015
G.f.: x*(3-2*x+3*x^2+x^3+4*x^4)/((1-x)*(1+x)*(1-x+x^2)). - Vincenzo Librandi, Mar 27 2015

More terms from Sascha Kurz, Mar 23 2002

Alternative to broken link added by R. J. Mathar, Jun 18 2010

A084407 Number of decimal places to which the n-th convergent of continued fraction expansion of Pi matches with the correct value.

Original entry on oeis.org

0, 2, 4, 6, 9, 9, 9, 9, 11, 10, 12, 12, 14, 15, 15, 16, 17, 17, 18, 19, 21, 23, 24, 24, 25, 27, 29, 30, 30, 32, 33, 34, 37, 39, 40, 40, 41, 42, 44, 45, 45, 46, 47, 49, 50, 51, 51, 53, 54, 55, 55, 56, 56, 58, 59, 59, 60, 60, 61, 60, 62, 64, 63, 64, 65, 65, 67, 67, 68, 70, 69, 71

Offset: 1

Lekraj Beedassy, Jun 24 2003

The n-th convergent of the continued fraction expansion of Pi is A002485(n+1)/A002486(n+1).

			From _A.H.M. Smeets_, Jun 13 2018: (Start)
Pi = 3.141592653589...
n=1: 3/1 = 3.0... so a(1) = 0;
n=2: 22/7 = 3.142... so a(2) = 2;
n=3: 333/106 = 3.14150... so a(3) = 4;
n=4: 355/113 = 3.1415929... so a(4) = 6;
n=5: 103993/33102 = 3.1415926530... so a(5) = 9;
n=6: 104348/33215 = 3.1415926539... so a(6) = 9;
n=7: 208341/66317 = 3.1415926534... so a(7) = 9;
n=8: 312689/99532 = 3.1415926536... so a(8) = 9;
n=9: 833719/265381 = 3.141592653581... so a(9) = 11;
n=10: 1146408/364913 = 3.14159265359... so a(10) = 10. (End)

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
F. Richman, Continued fractions

Cf. A000796.
Cf. A002485, A002486, A114526.
Cf. A086702, A100199, A240995.

Limit_{n -> oo} a(n)/n = 2*log(A086702)/log(10) = 2*A100199/log(10) = 2*A240995. - A.H.M. Smeets, Jun 13 2018

More terms from Vladeta Jovovic, Jun 27 2003

A068079 Decimal expansion of 355 / 113.

Original entry on oeis.org

3, 1, 4, 1, 5, 9, 2, 9, 2, 0, 3, 5, 3, 9, 8, 2, 3, 0, 0, 8, 8, 4, 9, 5, 5, 7, 5, 2, 2, 1, 2, 3, 8, 9, 3, 8, 0, 5, 3, 0, 9, 7, 3, 4, 5, 1, 3, 2, 7, 4, 3, 3, 6, 2, 8, 3, 1, 8, 5, 8, 4, 0, 7, 0, 7, 9, 6, 4, 6, 0, 1, 7, 6, 9, 9, 1, 1, 5, 0, 4, 4, 2, 4, 7, 7, 8, 7, 6, 1, 0, 6, 1, 9, 4, 6, 9, 0, 2, 6, 5, 4, 8, 6, 7, 2, 5, 6, 6, 3, 7, 1, 6, 8, 1, 4, 1, 5, 9, 2

Offset: 1

Nenad Radakovic, Mar 22 2002

This is an approximation to Pi. It is accurate to 0.00000849%.
355/113 is the third convergent of the continued fraction expansion of Pi (A001203). - Lekraj Beedassy, Jun 18 2003
In one of Ramanujan's papers, a note at the bottom states that "If the area of the circle be 140,000 square miles, then RD [RD = d/2 * Sqrt(355/113) = r*Sqrt(Pi), very nearly] is greater than the true length by about an inch."
This approximation of Pi was suggested by the astronomer Tsu Chúng-chih (A.D. 430 - 501) (see Gullberg). - Stefano Spezia, Jan 13 2025

			3.141592920353982300884955752212389380530973451327433628318584...

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 88.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 187, 238-239.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi, p. 91.
Ramanujan's papers, "Squaring the circle", Journal of the Indian Mathematical Society, V, 1913, 132. - Robert G. Wilson v, May 30 2014
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 49.

Dario Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163. [_N. J. A. Sloane_, Mar 24 2012]
Dale, Fun and interesting facts about Pi.
Index entries for sequences related to the number Pi.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1).

Cf. A068028, A068089, A002485, A002486, A046965, A046947, A083871.

Digits:=100: evalf(355/113); # Wesley Ivan Hurt, Mar 14 2015

Flatten[RealDigits[355/113, 10, 100]] (* Wesley Ivan Hurt, Mar 14 2015 *)

355/113. \\ Charles R Greathouse IV, May 30 2014

a(n) = if(n==1, 3, digits(16*10^112 \ 113)[(n-2) % 112 + 1]) \\ Jeppe Stig Nielsen, Dec 14 2019

a(n) = a(n - 112) for n > 113. - Jeppe Stig Nielsen, Dec 14 2019

More terms from Sascha Kurz, Mar 23 2002

Terms a(106) and beyond from Jeppe Stig Nielsen, Dec 14 2019

A138343 Count of post-period decimal digits up to which the rounded n-th convergent to Pi agrees with the exact value.

Original entry on oeis.org

0, 2, 3, 6, 8, 9, 8, 10, 10, 11, 11, 13, 15, 15, 16, 15, 17, 17, 18, 19, 20, 23, 24, 23, 26, 27, 29, 30, 29, 31, 33, 34, 37, 39, 39, 40, 42, 43, 44, 45, 45, 47, 46, 49, 49, 51, 52, 52, 54, 55, 56, 55, 56, 57, 59, 58, 59, 60, 61, 61, 63, 64, 64, 65, 65, 66, 67, 67, 68, 69, 70, 71, 72, 72

Offset: 0

Artur Jasinski, Mar 16 2008

This is a measure of the quality of the n-th convergent to A000796 if the convergent and the exact value are compared rounded to an increasing number of digits. (This is similar to A084407 which compares the truncated/floored values).
The sequence of rounded values of Pi is 3, 3.1, 3.14, 3.142, 3.1416, 3.14159, 3.141593, 3.1415927 etc, and the n-th convergent (provided by A002485 and A002486) is to be represented by its equivalent sequence.
a(n) represents the maximum number of post-period digits of the two sequences if compared at the same level of rounding. Counting only post-period digits (which is one less than the full number of decimal digits) is just a convention taken from A084407.

			For n=3, the 3rd convergent is 355/113 = 3.141592920353.., with a sequence of rounded representations 3, 3.1, 3.14, 3.142, 3.1416, 3.141593, 3.1415929, 3.14159292 etc.
Rounded to 1, 2, 3, 4, 5 or 6 post-period decimal digits, this is the same as the rounded version of the exact Pi, but disagrees if both are rounded to 7 decimal digits, where 3.1415927 <> 3.1415929.
So a(3) = 6 (digits), the maximum rounding level of agreement.

Cf. A138335, A138336, A138337, A138339.

Definition and values replaced as defined via continued fractions by R. J. Mathar, Oct 01 2009

A068089 Decimal expansion of 104348 / 33215.

Original entry on oeis.org

3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 9, 2, 1, 4, 2, 1, 0, 4, 4, 7, 0, 8, 7, 1, 5, 9, 4, 1, 5, 9, 2, 6, 5, 3, 9, 2, 1, 4, 2, 1, 0, 4, 4, 7, 0, 8, 7, 1, 5, 9, 4, 1, 5, 9, 2, 6, 5, 3, 9, 2, 1, 4, 2, 1, 0, 4, 4, 7, 0, 8, 7, 1, 5, 9, 4, 1, 5, 9, 2, 6, 5, 3, 9, 2, 1, 4, 2, 1, 0, 4, 4, 7, 0, 8, 7, 1, 5, 9, 4, 1, 5, 9, 2, 6, 5

Offset: 1

Nenad Radakovic, Mar 22 2002

This is an approximation to Pi. It is accurate to 0.00000001056%.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Dale, Facts about Pi
Index entries for sequences related to the number Pi
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A002485, A002486, A046965, A046947, A068028, A068079.

First[RealDigits[104348/33215,10,100]] (* Paolo Xausa, Nov 07 2023 *)

More terms from Sascha Kurz, Mar 23 2002

cp's OEIS Frontend

A136121 Integer Log of (Numerators convergent to Pi / Denominators convergent to Pi) = A001414(A002485/A002486) = A001414(A002485)-A001414(A002486).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A360370 Intersection of A002486 and A360367.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A356665 Number of correct decimal digits of the approximation of Pi obtained from the continued fraction convergents A002485(n)/A002486(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A002485 Numerators of convergents to Pi.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Extensions

A046947 Numbers k such that |sin(k)| (or |tan(k)| or |sec(k)|) decreases monotonically to 0; also |cos(k)| (or |cosec(k)| or |cot(k)|) increases.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A068028 Decimal expansion of 22/7.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A084407 Number of decimal places to which the n-th convergent of continued fraction expansion of Pi matches with the correct value.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A068079 Decimal expansion of 355 / 113.

Views

Author

Keywords