A068028 -id:A068028 - cp's OEIS frontend

A020806 Decimal expansion of 1/7.

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, 2

Offset: 0

N. J. A. Sloane

142857 and 999999 = 7*142857 are first and last Kaprekar numbers with six digits. Note a(n) + a(n+3) = 9. (142857^2 = 20408122449; 20408 + 122449 = 142857.) a(n)^2 = 1, 16, 4, 64, 25, 49, ... - Paul Curtz, Aug 24 2009
The constant 19 + 1/7 = 19.142857... is the Kirchhoff index of the Möbius ladder graph on v=8 vertices. The Laplacian matrix has the eigenvalues 4 (one time), 4-sqrt(2) (2 times), 4+sqrt(2) (2 times), 2 (2 times) and 0 (one time). Then the Kirchhoff index is v times the sum over the inverse, nonzero eigenvalues. - R. J. Mathar, Feb 13 2011
Decimal expansion of -99*(zeta(-5) + zeta(-9)) - 1. - Arkadiusz Wesolowski, Sep 15 2013
Also, decimal expansion of Sum_{i>0} 1/8^i. - Bruno Berselli, Jan 03 2014
The points whose coordinates are overlapping pairs of digits of this sequence, (1, 4), (4, 2), (2, 8), (8, 5), (5, 7) and (7, 1), all lie on one ellipse, with equation 19*x^2 + 36*x*y + 41*y^2 - 333*x - 531*y = -1638. Overlapping pairs of pairs of digits, (14, 28), (42, 85), (28, 57), (85, 71), (57, 14), (71, 42), also yield 6 points on one ellipse, with equation -165104*x^2 + 160804*x*y + 8385498*x - 41651*y^2 - 3836349*y = 7999600. (See book by Wells and MathWorld link.) - M. F. Hasler, Oct 25 2017

			0.142857142857142857...

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrüche'.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.

J. Hall, One-Seventh Ellipse, on MathWorld, by E. W. Weisstein.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A002371, A028416, A049084, A068028.

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

Digits:=100: evalf(1/7); # Wesley Ivan Hurt, Jun 28 2016

CoefficientList[Series[(1 + 3 x - 2 x^2 + 7 x^3) / ((1 - x) (1 + x) (1 - x + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 27 2015 *)
realDigitsRecip[7] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Sep 18 2024 *)

1/7. \\ Charles R Greathouse IV, Sep 24 2015

digits(10^99\7) \\ M. F. Hasler, Oct 25 2017

From  Reinhard Zumkeller, Oct 06 2008: (Start)
A028416(1)=7; A002371(A049084(7)) = A002371(4) = 6.
a(n+6) = a(n), a(n+6/2) = 9 - a(n). (End)
From Colin Barker, Aug 14 2012: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
G.f.: (1+3*x-2*x^2+7*x^3) / ((1-x)*(1+x)*(1-x+x^2)). (End)
a(n) = A068028(n+2). - Zak Seidov, Mar 26 2015
a(n) = (27 - 11*cos(n*Pi) - 10*cos(n*Pi/3) - 6*sqrt(3)*sin(n*Pi/3))/6. - Wesley Ivan Hurt, Jun 28 2016
E.g.f.: (8*cosh(x) - exp(x/2)*(5*cos(sqrt(3)*x/2) + 3*sqrt(3)*sin(sqrt(3)*x/2)) + 19*sinh(x))/3. - Stefano Spezia, Dec 07 2024

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

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

A003077 Decimal expansion of 22/7 - Pi.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.001264489267349618680213759577639972945687743482037036167912550549326450856...

Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 15.

G. C. Greubel, Table of n, a(n) for n = 0..10000
MathOverflow, Source and context of 22/7-Pi = Integral_{0..1} (x-x^2)^4/(1+x^2) dx ?
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (1.7.1)
Anthony Sofo, Euler related binomial sums, Indian J. Pure Appl. Math. 50 (1) (2019) 149-160
Eric Weisstein's MathWorld, Pi
Wikipedia, Pi
Index entries for transcendental numbers

Cf. A000796, A068028.

m:=152; [0,0] cat Reverse(Intseq(Floor(10^m*(22/7 - Pi(RealField(m+5)))))); // G. C. Greubel, Nov 03 2022

Join[{0, 0}, RealDigits[22/7-Pi, 10, 98][[1]]] (* Jean-François Alcover, May 24 2013 *)

22/7-Pi \\ Charles R Greathouse IV, Sep 30 2022

(22/7 - pi).n(digits=150).nextbelow() # G. C. Greubel, Nov 03 2022

22/7 - Pi = Integral_{x=0..1} x^4*(1-x)^4/(1+x^2). - M. F. Hasler, Oct 24 2011

Equals 60*Sum_{n>=1} 1/[(4*n^2-1)*(16*n^2-1)*(16*n^2-9)] . [Sofo] - R. J. Mathar, Jun 21 2024

A364844 a(n) is the n-digit numerator of the fraction h/k with h and k coprime palindrome positive integers at which abs(h/k-Pi) is minimal.

Original entry on oeis.org

3, 22, 474, 1551, 36163, 292292, 7327237, 31311313

Offset: 1

Stefano Spezia, Aug 10 2023

a(2) = 22 corresponds to the numerator of A068028.

			  n              fraction    approximated value
  -   -------------------    ------------------
  1                     3    3
  2                  22/7    3.1428571428571...
  3               474/151    3.1390728476821...
  4              1551/494    3.1396761133603...
  5           36163/11511    3.1416036834332...
  6          292292/93039    3.1416072829673...
  7       7327237/2332332    3.1415926206046...
  8      31311313/9966699    3.1415931192464...
  ...

Index entries for sequences related to the number Pi

Cf. A000796, A002113, A068028, A070252, A364845 (denominator), A364846.

Cf. A355622, A355623.

nmax = 8; a = {3}; hmin = kmin = 0; For[n = 2, n <= nmax, n++, minim = Infinity; h = Select[Range[10^(n - 1), 10^n - 1], PalindromeQ]; k = Select[Range[10^(n - 2), 10^n - 1], PalindromeQ]; lh = Length[h]; lk = Length[k]; For[i = 1, i <= lh, i++, For[j = 1, j <= lk, j++, If[(dist = Abs[Part[h, i]/Part[k, j] - Pi]) < minim && GCD[Part[h, i], Part[k, j]] == 1, minim = dist; hmin = Part[h, i]]]]; AppendTo[a, hmin]]; a

A364845 a(n) is the denominator of the fraction h/k with h and k coprime palindrome positive integers at which abs(h/k-Pi) is minimal, with the numerator h of n digits.

Original entry on oeis.org

1, 7, 151, 494, 11511, 93039, 2332332, 9966699

Offset: 1

Stefano Spezia, Aug 10 2023

a(2) = 7 corresponds to the denominator of A068028.

			  n              fraction    approximated value
  -   -------------------    ------------------
  1                     3    3
  2                  22/7    3.1428571428571...
  3               474/151    3.1390728476821...
  4              1551/494    3.1396761133603...
  5           36163/11511    3.1416036834332...
  6          292292/93039    3.1416072829673...
  7       7327237/2332332    3.1415926206046...
  8      31311313/9966699    3.1415931192464...
  ...

Index entries for sequences related to the number Pi

Cf. A000796, A002113, A068028, A070252, A364844 (numerator), A364846.

Cf. A355622, A355623.

nmax = 8; a = {1}; hmin = kmin = 0; For[n = 2, n <= nmax, n++, minim = Infinity; h = Select[Range[10^(n - 1), 10^n - 1], PalindromeQ]; k = Select[Range[10^(n - 2), 10^n - 1], PalindromeQ]; lh = Length[h]; lk = Length[k];  For[i = 1, i <= lh, i++, For[j = 1, j <= lk, j++, If[(dist = Abs[Part[h, i]/Part[k, j] - Pi]) < minim && GCD[Part[h, i], Part[k, j]] == 1, minim = dist; kmin = Part[k, j]]]]; AppendTo[a, kmin]]; a

A210621 Decimal expansion of 256/81.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 24 2012

According to Maor (1994), the Rhind Papyrus asserts that a circle has the same area as a square with a side that is 8/9 the diameter of the circle. From this we can determine that 256/81 is one of the ancient Egyptian approximations of Pi. - Alonso del Arte, Jun 12 2012

			3.1604938271604938271604938271604938271604938271604938271604...

Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 12.
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 p. 237.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi, p. 89.
Carl Theodore Heisel, Behold! The grand problem no longer unsolved: The circle squared beyond refutation, c. 1935. (proposes Pi = 3 + 13/81)
Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994): 41, 47 note 1.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 48.

Dario Castellanos, The ubiquitous Pi, Math. Mag., 61 (1988), 67-98 and 148-163.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A068028, A229943, A255910, A268315.

RealDigits[256/81, 10, 100][[1]] (* Alonso del Arte, Jun 12 2012 *)

256/81. \\ Charles R Greathouse IV, Sep 13 2013

256/81 = (4/3)^4.

Equals 3*A229943 = A255910^2 = A268315/3. - Hugo Pfoertner, Jun 26 2024

Offset corrected by Rick L. Shepherd, Jan 06 2014

A216606 Decimal expansion of 360/7.

Original entry on oeis.org

5, 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

Offset: 2

Paul Curtz, Sep 10 2012

A020806 preceded by a 5.

Number of degrees in the exterior angle of an equilateral heptagon. Since 1969, used in many (orbiform or Reuleaux) heptagonal coins. Zambia has a natural heptagonal coin. Brazil and Costa Rica have a coin with the natural heptagon inscribed in the coin's disk.

			51.42857...

World of Coins, An Alphabet of Heptagons: Seven-sided Coins.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A019695, A020806, A021018, A060708, A068028, A178817, A178818.

Digits:=100: evalf(360/7); # Wesley Ivan Hurt, Jun 28 2016

Flatten[RealDigits[360/7, 10, 100]] (* Wesley Ivan Hurt, Jun 28 2016 *)

360/7. \\ Charles R Greathouse IV, Sep 12 2012

a(n) = 50 + 10*A020806(n).
After 5, of period 6: repeat [1, 4, 2, 8, 5, 7].
From Wesley Ivan Hurt, Jun 28 2016: (Start)
G.f.: x^3*(5-4*x+3*x^2+3*x^3+2*x^4) / (1-x+x^3-x^4).
a(n) = 9/2 + 11*cos(n*Pi)/6 + 5*cos(n*Pi/3)/3 + sqrt(3)*sin(n*Pi/3), n>2.
a(n) = a(n-1) - a(n-3) + a(n-4) for n>6, a(n) = a(n-6) for n>8. (End)

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).

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).

cp's OEIS Frontend

A020806 Decimal expansion of 1/7.

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A068079 Decimal expansion of 355 / 113.

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A068089 Decimal expansion of 104348 / 33215.

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A003077 Decimal expansion of 22/7 - Pi.

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A364844 a(n) is the n-digit numerator of the fraction h/k with h and k coprime palindrome positive integers at which abs(h/k-Pi) is minimal.

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A364845 a(n) is the denominator of the fraction h/k with h and k coprime palindrome positive integers at which abs(h/k-Pi) is minimal, with the numerator h of n digits.

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A210621 Decimal expansion of 256/81.

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