has decimal expansion given by
(1) 
(OEIS A000796). The following table summarizes some record computations of the digits of .
1999  Kanada, Ushio and Kuroda 
Dec. 2002  Kanada, Ushio and Kuroda (Peterson 2002, Kanada 2003) 
Aug. 2012  A. J. Yee (Yee) 
Aug. 2012  S. Kondo and A. J. Yee (Yee) 
Dec. 2013  A. J. Yee and S. Kondo (Yee) 
The calculation of the digits of has occupied mathematicians since the day of the Rhind papyrus
(1500 BC). Ludolph van Ceulen spent much of his life calculating to 35 places. Although he did not live to publish his result,
it was inscribed on his gravestone. Wells (1986, p. 48) discusses a number of
other calculations. The calculation of also figures in the Season 2 Star
Trek episode “Wolf
in the Fold” (1967), in which Captain Kirk and Mr. Spock force an evil
entity (composed of pure energy and which feeds on fear) out of the starship Enterprise’s
computer by commanding the computer to “compute to the last digit the value
of pi,” thus sending the computer into an infinite loop.
AlKāshi of Samarkand computed the sexagesimal digits of
as
(2) 
(OEIS A091649) using gons, a value accurate to 17 decimal places (Borwein
and Bailey 2003, p. 107).
The binary representation of the decimal digits of (top figure) and decimal representation (bottom figure) of
are illustrated above.
A plot of the first 1600 decimal digits of (mod 2) is shown above (left figure), with the corresponding
plot for 22/7 shown at right. Here, white indicates an even digit and black an odd
digit (Pickover 2002, p. 285).
Spigot (Rabinowitz and Wagon 1995; Arndt and Haenel 2001; Borwein and Bailey 2003, pp. 140141) and base16 digitextraction
algorithms (the BBP formula) are known for . A remarkable recursive formula conjectured
to give the th
hexadecimal digit of is given by , where is the floor function,
(3) 
is the fractional
part and
(Borwein and Bailey 2003, Ch. 4; Bailey et al. 2007, pp. 2223).
The limit pi formulas
(4) 
and
(5) 
where is a Bernoulli
number (Plouffe 2022) can be used as a digitextraction
algorithm for
(as well as ).
In particular, letting
(6) 
the th digit to the right of the decimal point
of for is given by
(7) 
where is the integer
part and
is the fractional part. Similar formulas can be
obtained using
(8) 
and
(9) 
where is an Euler
number, which gives a base9 (or binary) digitextraction
algorithm (Plouffe 2022). Related limits and formulas can also be obtained for
(Plouffe 2022).
Piprimes, i.e., constant primes occur
at 2, 6, 38, 16208, 47577, 78073, 613373, … (OEIS A060421)
decimal digits.
The beast number 666 appears in at decimals 2440, 3151, 4000, 4435, 5403, 6840, (OEIS A083625). The first occurrences of just consecutive 6s are 7, 117, 2440, 21880, 48439, 252499, 8209165,
55616210, 45681781, … (OEIS A096760), while
(or more) consecutive 6s first occur
at 7, 117, 2440, 21880, 48439, 252499, 8209165, 45681781, 45681781, … (OEIS A050285).
The digits 314159 appear at positions 176451, 1259351, 1761051, 6467324, 6518294, 9753731, 9973760, … (correcting Pickover 1995).
The sequence 0123456789 occurs beginning at digits , , , , , and (OEIS A101815;
cf. Wells 1986, pp. 5152).
The sequence 9876543210 occurs beginning at digits , , , , and (OEIS A101816).
The sequence 27182818284 (the first few digits of e) occurs beginning at digit
(see also Pickover’s sequence).
There are also interesting patterns for . 0123456789 occurs at , 9876543210 occurs at and , and 999999999999 occurs at of .
The starting positions of the first occurrence of , 1, 2, … in the decimal expansion of (including the initial 3 and counting it as the first digit)
are 33, 2, 7, 1, 3, 5, 8, 14, … (OEIS A032445).
Scanning the decimal expansion of until all digit numbers have occurred, the last 1, 2, … digit numbers
appearing are 0, 68, 483, 6716, 33394, 569540, … (OEIS A032510),
which end at digits 33, 607, 8556, 99850, 1369565, … (OEIS A080597).
A curiosity relating
to the beast number 666 involves adding the first
three sextads of .
First, note that
(10) 
Now, skip ahead 15 decimal places and note that the sum is repeated as
(11) 
(pers. comm., P. Olivera, Aug. 11, 2005; Olivera).
It is not known if
is normal (Wagon 1985, Bailey and Crandall 2001),
although the first 30 million digits are very uniformly
distributed (Bailey 1988).
The following distribution of decimal digits is found for the first digits of (Kanada 2003). It shows no statistically significant
departure from a uniform distribution.
OEIS  2  3  4  5  6  7  8  9  10  11  12  
0  A099291  8  93  968  9999  99959  999440  9999922  99993942  999967995  10000104750  99999485134 
1  A099292  8  116  1026  10137  99758  999333  10002475  99997334  1000037790  9999937631  99999945664 
2  A099293  12  103  1021  9908  100026  1000306  10001092  100002410  1000017271  10000026432  100000480057 
3  A099294  11  102  974  10025  100229  999964  9998442  99986911  999976483  9999912396  99999787805 
4  A099295  10  93  1012  9971  100230  1001093  10003863  100011958  999937688  10000032702  100000357857 
5  A099296  8  97  1046  10026  100359  1000466  9993478  99998885  1000007928  9999963661  99999671008 
6  A099297  9  94  1021  10029  99548  999337  9999417  100010387  999985731  9999824088  99999807503 
7  A099298  8  95  970  10025  99800  1000207  9999610  99996061  1000041330  10000084530  99999818723 
8  A099299  12  101  948  9978  99985  999814  10002180  100001839  999991772  10000157175  100000791469 
9  A099300  14  106  1014  9902  100106  1000040  9999521  100000273  1000036012  9999956635  99999854780 
The following table gives the first few positions at which a digit occurs times. The sequence 1, 135, 1698, 54525, 24466, 252499, 3346228,
46663520, 564665206, … (OEIS A061073) given
by the diagonal (plus any terms of the form 10 10’s etc.) is known as the Earls
sequence (Pickover 2002, p. 339). The sequence 999999 occurs at decimal
762 (which is sometimes called the Feynman point;
Wells 1986, p. 51) and continues as 9999998, which is largest value of any seven
digits in the first million decimals.
OEIS  strings of 1, 2, … 

0  A050279  32, 307, 601, 13390, 17534, 1699927, … 
1  A035117  1, 94, 153, 12700, 32788, 255945, … 
2  A050281  6, 135, 1735, 4902, 65260, 963024, … 
3  A050282  9, 24, 1698, 28467, 28467, 710100, … 
4  A050283  2, 59, 2707, 54525, 808650, 828499, … 
5  A050284  4, 130, 177, 24466, 24466, 244453, … 
6  A050285  7, 117, 2440, 21880, 48439, 252499, … 
7  A050286  13, 559, 1589, 1589, 162248, 399579, … 
8  A050287  11, 34, 4751, 4751, 213245, 222299, … 
9  A048940  5, 44, 762, 762, 762, 762, 1722776, … 