The pi continued fraction is the expansion of pi as a simple continued fraction, 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...](/images/equations/PiContinuedFraction/NumberedEquation1.svg) |
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is an 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 convergent 
of 
is given by
![pi approx 2[1;1,1,3,32]=(355)/(113) approx 3.14159292...](/images/equations/PiContinuedFraction/NumberedEquation2.svg) |
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(Stoschek, undated).
The Engel expansion of 
is 1, 1, 1, 8, 8, 17, 19, 300, 1991, 2492, ... (OEIS A006784).
The following table summarizes historical record computations through 2013 of the continued fraction of pi.
| terms | date | reference |
 | 1977 | W. Gosper (Gosper 1977, Ball and Coxeter 1987) |
 | Jun. 1999 | H. Havermann (Plouffe 2000) |
 | 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 |
 | Apr. 27, 2021 | S. Fahad (Fahad 2021) |
 | Nov. 1, 2025 | M. Frank (Frank 2025) |
The positions of the first occurrence of 
, 2, ... in the continued
fraction, counting the initial 3 as position 0, are 3, 8, 0, 29, 39, 31, 1, 43,
129, 99, ... (OEIS A225802). The smallest integers which do not occur in the first 
terms are 49004, 50471, 53486, 56315, ... (E. Weisstein,
Jul. 27, 2013). The sequence of successive record
partial quotients 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), here counting the initial
3 as position 1. The largest term among the first 653520000000 terms computed by
Frank (2025) is 845188380288.
Let the continued fraction of 
be denoted ![[a_0;a_1,a_2,...]](/images/equations/PiContinuedFraction/Inline19.svg)
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 to 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, 12051034, 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
 |
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(Khrushchev 2006), obtained by Brouncker in 1655 and giving convergents 1, 3/2, 15/13, 105/76, 315/263, ... (OEIS A025547 and A007509) and
 |
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(Stern 1833, pp. 267-268), giving convergents 1, 2/3, 4/3, 16/15, 64/45, 128/105, ... (OEIS A001901 and A046126).
."
Math. Comput. 25, 387-392, 1971. 
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." 
." Monatsh. für Math. 67, 311-316,
1963.Plouffe, S. "Table of Current Records for the Computation
of Constants." Jun. 8, 2000.