Pi Continued Fraction

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

(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  is given by

(2)

(Stoschek).

The Engel expansion of  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.

terms	date	reference
	1977	W. Gosper (Gosper 1977, Ball and Coxeter 1987)
	Jun. 1999	H. Havermann (Plouffe)
	Mar. 2002	H. Havermann (Bickford)
	Oct. 2010	N. Bickford (Bickford 2010, Wolfram Blog Team 2011)
	Dec. 2010	E. W. Weisstein
	Sep. 16, 2011	E. W. Weisstein
	Sep. 17, 2011	E. W. Weisstein
	Sep. 18, 2011	E. W. Weisstein
	Jul. 18, 2013	E. W. Weisstein
	Jul. 27, 2013	E. W. Weisstein

The positions of the first occurrence of , 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  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)

Let the continued fraction of  be denoted  and let the denominators of the convergents be denoted , , ..., . Then plots above show successive values of , , , which appear to converge to Khinchin's constant (left figure) and , 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 -digit terms in the continued fraction of , counting 3 as the 0th (e.g., Choong et al. 1971, Beeler et al. 1972).

	OEIS	terms/positions
1	A048292	3, 7, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, ...
	A048293	0, 1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, ...
2	A048294	15, 14, 84, 15, 13, 99, 12, 16, 45, 22, ...
	A048955	2, 12, 21, 25, 27, 33, 54, 77, 80, 82, ...
3	A048956	292, 161, 120, 127, 436, 106, 141, ...
	A048957	4, 79, 196, 222, 307, 601, 669, 725, ...
4	A048958	1722, 2159, 8277, 1431, 1282, 2050, ...
	A048959	3273, 3777, 3811, 4019, 4700, 6209, ...
5	A048960	20776, 19055, 19308, 78629, 17538, ...
	A048961	431, 15543, 23398, 28421, 51839, ...
6		179136, 528210, 104293, 196030, ...
		156381, 267313, 294467, 513205, ...
7		8093211, 1811791, 3578547, 4506503, ...
		1118727, 2782369, 2899883, 3014261, ...
8		12996958 ,19626118, 12051Q034, 13435395, ...
		453293, 27741604, 46924606, 50964645, ...
9		878783625, 317579569, ...
		11504930, 74130513, ...

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

(3)

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

(4)

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

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  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 ." Monatsh. für Math. 67, 311-316, 1963.

Sloane, N. J. A. Sequences 0012032646,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.