A068079 -id:A068079 - 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

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

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).

A083871 Decimal expansion of sqrt(355/113).

Original entry on oeis.org

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

Offset: 1

Lekraj Beedassy, Jun 18 2003

355/113 is an approximation to Pi, accurate to the first 7 digits. - Harvey P. Dale, Apr 22 2022

			1.772453926158302796091946476063677662004305549....

S. Ramanujan, Fig.1:Geometrical construction for sqrt(355/113)
S. Ramanujan, Squaring The Circle
J. C. Santos, Quadrature of the circle
Wikipedia, Squaring the circle
G. Xiao, Numerical Calculator, Operate on "sqrt(355/133)"

Cf. A068079, A002161, A002485.

RealDigits[Sqrt[355/113], 10, 106][[1]] (* Georg Fischer, Apr 03 2020 *)

9 inserted behind a(17) and a(71) corrected by Georg Fischer, Apr 03 2020

A224365 a(n) = A063674(n+1) - A063674(n).

Original entry on oeis.org

10, 3, 3, 3, 157, 22, 22, 22, 22, 22, 22, 22, 22, 51808, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355

Offset: 1

Paul Curtz, Apr 09 2013

The repeated terms (3, 22, 355, 5419351, ... from A063674) are the numerators of fractions (3/1, 22/7, 355/113, 5419351/1725033, ...) leading to Pi.
Zu Chongzhi (5th century) discovered 22/7 and 355/113. Adriaan Anthonisz Metius rediscovered 355/113 in 1585.
First differences of A063673 give the denominators: 3, 1, 1, 1, 50, 7, 7, 7, 7, 7, 7, 7, 7, 16489, 113, 113, ... .
Hence 10/3, 157/50, 51808/16489, ... .

Vincenzo Librandi, Table of n, a(n) for n = 1..168

Cf. A063673, A063674.

Cf. A046947, A072398, A002485. A132049, A003077, A068028, A068079.

A224365 = Reap[ For[ delta0 = 1; d = 1, d < 10^5, d++, delta = Abs[Pi - Round[Pi*d]/d]; If[ delta < delta0, Sow[ Round[Pi*d]]; delta0 = delta]]][[2, 1]] // Differences (* Jean-François Alcover, Apr 10 2013 *)

a(n) = A063674(n+1) - A063674(n).

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

A375820 Decimal expansion of 333/106.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Aug 30 2024

Approximation of Pi accurate up to five digits.

Periodic of period [4, 1, 5, 0, 9, 4, 3, 3, 9, 6, 2, 2, 6] of length 13.

			3.1415094339622641509433962264150943396226415...

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 154-155.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 49.

Michael A. B. Deakin and H. Lausch, The Bible and Pi, The Mathematical Gazette, 82(494), 162-166 (1998).
Wikipedia, Adriaan Metius.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000796, A068079, A210622.

Mathematica
```
RealDigits[333/106,10,100][[1]]
```

def A375820(n): return (2, 2 ,6, 4, 1, 5, 0, 9, 4, 3, 3, 9, 6)[n%13] if n>2 else 5-(n<<1) # Chai Wah Wu, Aug 30 2024

Equals (2*355 - 377)/(2*113 - 120), where A068079 = 355/113 and A210622 = 377/120 [Adriaen Metius]. See Wells.

A021117 Decimal expansion of 1/113.

Original entry on oeis.org

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, 9, 2, 0, 3, 5, 3, 9, 8, 2, 3

Offset: 0

N. J. A. Sloane

Periodic with period length 112. - Ray Chandler, Jan 23 2024

			0.0088495575221238938...

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. A001913, A068079.

evalf(1/113, 110); # Wesley Ivan Hurt, Feb 13 2017

Join[{0,0},RealDigits[1/113,10,120][[1]]] (* Harvey P. Dale, Nov 16 2021 *)

Extended to a full period by Sean A. Irvine, May 07 2019

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

cp's OEIS Frontend

A068028 Decimal expansion of 22/7.

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A068089 Decimal expansion of 104348 / 33215.

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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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A083871 Decimal expansion of sqrt(355/113).

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A224365 a(n) = A063674(n+1) - A063674(n).

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

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A375820 Decimal expansion of 333/106.

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