A068089 -id:A068089 - cp's OEIS frontend

A068028 Decimal expansion of 22/7.

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

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

A072398 Numerator of best approximation to Pi with denominator <= 10^n.

Original entry on oeis.org

3, 22, 22, 355, 355, 312689, 1146408, 5419351, 245850922, 2549491779, 21053343141, 21053343141, 1783366216531, 8958937768937, 139755218526789, 428224593349304, 30246273033735921, 66627445592888887

Offset: 0

Rick L. Shepherd, Jun 15 2002

			A072398(5) = 312689 because A072398(5)/A072399(5) = 312689/99532 is the best rational approximation to Pi with positive denominator <= 10^5 = 100000. This approximation is accurate to 0.00000000092766%.

Daniel Mondot, Table of n, a(n) for n = 0..999

Cf. A072399 (denominators), A000796 (Pi), A068089, A002485/A002486.

nmax = 17; cv = Convergents[Pi, 2*nmax] // Reverse; a[n_] := Select[cv, Denominator[#] <= 10^n &, 1] // Numerator // First; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jan 04 2013 *)

for(n=0,40,print1(numerator(bestappr(Pi,10^n)),",")) \\ Finds these approximations very quickly.

A072399 Denominator of best approximation to Pi with denominator <= 10^n.

Original entry on oeis.org

1, 7, 7, 113, 113, 99532, 364913, 1725033, 78256779, 811528438, 6701487259, 6701487259, 567663097408, 2851718461558, 44485467702853, 136308121570117, 9627687726852338, 21208174623389167, 842468587426513207

Offset: 0

Rick L. Shepherd, Jun 15 2002

			a(6) = 364913 because A072398(6)/a(6) = 1146408/364913 is the best rational approximation to Pi with positive denominator <= 10^6 = 1000000. This approximation is accurate to 0.000000000051271%.

Daniel Mondot, Table of n, a(n) for n = 0..999

Cf. A072398 (numerators), A000796 (Pi), A068089, A002485/A002486.

nmax = 18; cv = Convergents[Pi, 2*nmax] // Reverse; a[n_] := Select[cv, Denominator[#] <= 10^n &, 1] // Denominator // First; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jan 04 2013 *)

for(n=0,40,print1(denominator(bestappr(Pi,10^n)),",")) \\ Finds these approximations very quickly.

A328927 Decimal expansion of (9^2 + (19^2)/22)^(1/4): an approximation for Pi from Srinivasa Ramanujan.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Oct 31 2019

Srinivasa Ramanujan published this curious empirical approximation in 1914 accompanied with a simple geometric construction for Pi based on this value of (9^2 + (19^2)/22)^(1/4) [See Ramanujan link, page 43, section 12, and page 44, Figure 2].
S. Ramanujan found 3.14159265262... as the value for this approximation in 1914 while Maple gives 3.14159265258... and Pi = 3.14159265358...
This approximation is correct to 10^(-8).
Gardner (1985) wrote: "A more astounding discovery is that: 22 pi^4 = 2143. A few multiplications, and the 10 million-plus decimals of pi have vanished. (Can this remarkable relationship mirror some as yet undiscovered facet of physical reality?)" In the Postscript to the 1999 reprint he writes "Divide 2143 (the first four counting numbers) by 22 and hit the square-root button twice. You will get pi to eight decimals", and credits this discovery to Srinivasa Ramanujan. The MathOverflow page also mentions this and the near-integer 10*Pi^4 - 1/11 ≈ 974.0000012... See A352548 for 22*Pi^4. - M. F. Hasler, Jun 22 2022

			3.141592652582646125206037179644022371557877983160126149695135327918621058849...

Jörg Arndt and Christoph Haenel, Pi Unleashed, Springer-Verlag, 2006, retrieved 5 June 2013, (4.18), page 58.
Martin Gardner, "Slicing Pi into Millions", Discover, 6:50, January 1985.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 3.14159 (Pi), page 36.

William R. Corliss, The Secret Of It All Is In The Pi, Science Frontiers #37, Jan-Feb 1985.
Martin Gardner, Slicing Pi into Millions, in: Gardner's Why and Wherefores, Prometheus Books (1999), p.87.
Srinivasa Ramanujan, Modular equations and approximations to Pi, Quarterly Journal of Mathematics, XLV, 1914, 43-44.
Zurab Silagadze, The origin of the Ramanujan's π^4 ≈ 2143/22 identity, MathOverflow.net, Feb 26 2016.
Wikipedia, Approximations of π: Miscellaneous_approximations.

Cf. A000796, A068028, A068079, A068089, A352548.

Maple
```
evalf((9^2 + (19^2)/22)^(1/4),125);
```

RealDigits[Surd[9^2 + (19^2)/22, 4], 10, 120][[1]] (* Amiram Eldar, Jun 18 2023 *)

A328927_first(N)=localprec(N+9); digits(10^N\sqrtn(22/.2143,4)) \\ First N terms of the sequence, i.e., a(1, 0, -1, ..., 2-N). - M. F. Hasler, Jun 22 2022

Equals (102 - 2222/(22^2))^(1/4) = (2143/22)^(1/4).

A374322 Decimal expansion of sqrt(2)*9801/4412.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 04 2024

Approximates Pi, correct to 7 digits.

			3.1415927300133056603139961890252155185995816071100335596565...

Paolo Xausa, Table of n, a(n) for n = 1..10000
David H. Bailey and Jonathan M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the American Mathematical Society, May 2005, Vol. 52, No. 5, p. 511.
Jonathan M. Borwein, The Experimental Mathematician: the Pleasure of Discovery and the Role of Proof, International Journal of Computers for Mathematical Learning, Vol. 10 (2005), p. 97.
Index entries for sequences related to the number Pi

Cf. A000796, A002193, A068028, A068079, A068089, A253214.

First[RealDigits[Sqrt[2]*9801/4412, 10, 100]]

from math import isqrt
def A374322(n): return isqrt(10**(n-1<<1)*96059601//9732872)%10 # Chai Wah Wu, Jul 04 2024

A339264 Decimal expansion of (63/25) * (17+15sqrt(5)) / (7+15sqrt(5)): an approximation for Pi from Srinivasa Ramanujan.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Nov 29 2020

This formula that derives from Ramanujan modular equations is correct to 9 places exactly (see Ramanujan link, page 43).
Pi = 3.1415926535... and this approximation = 3.1415926538...
A quadratic number with minimal polynomial 168125x^2 - 792225x + 829521 and denominator 6725. - Charles R Greathouse IV, Oct 02 2022

			3.141592653805688201898390006301507822487503475774...

Jörg Arndt and Christoph Haenel, Pi Unleashed, Springer-Verlag, 2006, retrieved Jun 05 2013, (4.17) page 57.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 3.14159 (Pi), page 36.

S. Ramanujan, Modular equations and approximations to Pi, Quarterly Journal of Mathematics, XLV, 1914, p. 43.
Index entries for algebraic numbers, degree 2

Other approximations to Pi: A068028, A068079, A068089, A328927.

evalf((63/25)*(17+15*sqrt(5))/(7+15*sqrt(5)),100);

RealDigits[(63/25)*(17 + 15*Sqrt[5])/(7 + 15*Sqrt[5]), 10, 100][[1]] (* Amiram Eldar, Nov 29 2020 *)

(63/13450) * (503+75*sqrt(5)) \\ Michel Marcus, Nov 29 2020

Equals (63/13450) * (503+75*sqrt(5)).

Equals the root of 829521 - 792225*x + 168125*x^2 which is > 3. - Peter Luschny, Nov 29 2020

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A068028 Decimal expansion of 22/7.

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A068079 Decimal expansion of 355 / 113.

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A072398 Numerator of best approximation to Pi with denominator <= 10^n.

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A072399 Denominator of best approximation to Pi with denominator <= 10^n.

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A328927 Decimal expansion of (9^2 + (19^2)/22)^(1/4): an approximation for Pi from Srinivasa Ramanujan.

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A374322 Decimal expansion of sqrt(2)*9801/4412.

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A339264 Decimal expansion of (63/25) * (17+15*sqrt(5)) / (7+15*sqrt(5)): an approximation for Pi from Srinivasa Ramanujan.

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A339264 Decimal expansion of (63/25) * (17+15sqrt(5)) / (7+15sqrt(5)): an approximation for Pi from Srinivasa Ramanujan.