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Pi Continued Fraction

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pi continued fraction binary plot

The simple continued fraction for pi is given by [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] (OEIS A001203). A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.

The first few convergents are 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively.

The very large term 292 means that the convergent

[3;7,15,1]=[3,7,16]=(355)/(113)=3.14159292...
(1)

is an extremely good approximation good to six decimal places that was first discovered by astronomer Tsu Ch'ung-Chih in the fifth century A.D. (Gardner 1966, pp. 91-102). A nice expression for the third convergent of pi is given by

pi approx 2[1;1,1,3,32]=(355)/(113) approx 3.14159292...
(2)

(Stoschek).

The Engel expansion of pi is 1, 1, 1, 8, 8, 17, 19, 300, 1991, 2492, ... (OEIS A006784).

The following table summarizes some record computations of the continued fraction of pi.

termsdatereference
170013031977W. Gosper (Gosper 1977, Ball and Coxeter 1987)
20000000Jun. 1999H. Havermann (Plouffe)
180×10^6Mar. 2002H. Havermann (Bickford)
458×10^6Oct. 2010N. Bickford (Bickford 2010, Wolfram Blog Team 2011)
1940535772Dec. 2010E. W. Weisstein
2910789567Sep. 16, 2011E. W. Weisstein
4851308496Sep. 17, 2011E. W. Weisstein
5821569425Sep. 18, 2011E. W. Weisstein
10672905501Jul. 18, 2013E. W. Weisstein
15000000000Jul. 27, 2013E. W. Weisstein

The positions of the first occurrence of n=1, 2, ... in the continued fraction are 3, 8, 0, 29, 39, 31, 1, 43, 129, 99, ... (OEIS A225802). The smallest integers which does not occur in the first 1.5×10^(10) terms are 49004, 50471, 53486, 56315, ... (E. Weisstein, Jul. 27, 2013). The sequence of increasing terms in the continued fraction is 3, 7, 15, 292, 436, 20776, 78629, 179136, 528210, 12996958, 878783625, 5408240597, 5916686112, 9448623833, ... (OEIS A033089), occurring at positions 1, 2, 3, 5, 308, 432, 28422, 156382, 267314, 453294, 11504931 ... (OEIS A033090).

PiKhinchinLevy

Let the continued fraction of pi be denoted [a_0;a_1,a_2,...] and let the denominators of the convergents be denoted q_1, q_2, ..., q_n. Then plots above show successive values of a_1^(1/1), (a_1a_2)^(1/2), (a_1a_2...a_n)^(1/n), which appear to converge to Khinchin's constant (left figure) and q_n^(1/n), which appear converge to the Lévy constant (right figure), although neither of these limits has been rigorously established.

The following table gives the first few occurrences of d-digit terms in the continued fraction of pi, counting 3 as the 0th (e.g., Choong et al. 1971, Beeler et al. 1972).

dOEISterms/positions
1A0482923, 7, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, ...
A0482930, 1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, ...
2A04829415, 14, 84, 15, 13, 99, 12, 16, 45, 22, ...
A0489552, 12, 21, 25, 27, 33, 54, 77, 80, 82, ...
3A048956292, 161, 120, 127, 436, 106, 141, ...
A0489574, 79, 196, 222, 307, 601, 669, 725, ...
4A0489581722, 2159, 8277, 1431, 1282, 2050, ...
A0489593273, 3777, 3811, 4019, 4700, 6209, ...
5A04896020776, 19055, 19308, 78629, 17538, ...
A048961431, 15543, 23398, 28421, 51839, ...
6179136, 528210, 104293, 196030, ...
156381, 267313, 294467, 513205, ...
78093211, 1811791, 3578547, 4506503, ...
1118727, 2782369, 2899883, 3014261, ...
812996958 ,19626118, 12051Q034, 13435395, ...
453293, 27741604, 46924606, 50964645, ...
9878783625, 317579569, ...
11504930, 74130513, ...

The simple continued fraction for pi does not show any obvious patterns, but clear patterns do emerge in the beautiful non-simple continued fractions

4/pi=1+(1^2)/(2+(3^2)/(2+(5^2)/(2+(7^2)/(2+...))))
(3)

(Brouncker), giving convergents 1, 3/2, 15/13, 105/76, 315/263, ... (OEIS A025547 and A007509) and

pi/2=1-1/(3-(2·3)/(1-(1·2)/(3-(4·5)/(1-(3·4)/(3-(6·7)/(1-(5·6)/(3-...)))))))
(4)

(Stern 1833), giving convergents 1, 2/3, 4/3, 16/15, 64/45, 128/105, ... (OEIS A001901 and A046126).

See also

Euler-Mascheroni Constant Continued Fraction, Pi, Pi Approximations, Pi Digits, Pi Formulas

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References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 55 and 274, 1987.Beeler, M. et al. Item 140 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 69, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/pi.html#item140.Bickford, N. "Pi." http://nbickford.wordpress.com/2010/10/22/pi/. Oct. 22, 2010.Choong, Daykin, and Rathbone. Math. Comput. 25, 387, 1971.Gardner, M. "The Transcendental Number Pi." Ch. 8 in Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 91-102, 1966.Gosper, R. W. Table of Simple Continued Fraction for pi and the Derived Decimal Approximation. Stanford, CA: Artificial Intelligence Laboratory, Stanford University, Oct. 1975. Reviewed in Math. Comput. 31, 1044, 1977.Havermann, H. "Simple Continued Fraction Expansion of Pi." http://odo.ca/~haha/cfpi.html.Lochs, G. "Die ersten 968 Kettenbruchnenner von pi." Monatsh. für Math. 67, 311-316, 1963.Sloane, N. J. A. Sequences A001203/M2646,A002485/M3097, A002486/M4456, A114526, and A225802 in "The On-Line Encyclopedia of Integer Sequences."Stoschek, E. "Modul 33: Algames with Numbers." http://marvin.sn.schule.de/~inftreff/modul33/task33.htm.Wolfram Blog Team. "From Pi to Puzzles." http://blog.wolfram.com/2011/09/15/from-pi-to-puzzles/. Sep. 15, 2011.

Cite this as:

Weisstein, Eric W. "Pi Continued Fraction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PiContinuedFraction.html

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