Pythagoras's Constant Digits
Pythagoras's constant 
has decimal expansion
(OEIS A000129), It was computed to 
decimal digits by A. J. Yee on Feb. 9, 2012.
The Earls sequence (starting position of 
copies of the digit 
) for 
is given for 
, 2, ... by 2, 114, 1481, 3308, 72459, 226697, 969836, 119555442, 2971094743, ... (OEIS A224871).
-constant primes occur at 55, 97, 225, 11260, 11540, ... (OEIS A115377) decimal digits.
The starting positions of the first occurrence of 
, 1, 2, ... in the decimal expansion of 
(including the initial 1 and counting it as the first digit) are 14, 1, 5, 7, 2, 8, 9, 12, 19, ... (OEIS A229199).
Scanning the decimal expansion of 
until all 
-digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 8, 81, 748, 8505, 30103, 489568, ... (OEIS A000000), which end at digits 19, 420, 8326, 94388, 1256460, 13043524, ... (OEIS A000000).
The digit sequence 9876543210 does not occur in the first 
digits of 
, but 0123456789 does, starting at positions 864106288, 6458611884, 7311432557, ... (OEIS A000000) (E. Weisstein, Jul. 22, 2013).
It is not known if 
is normal (Beyer et al. 1969, 1970ab), but the following table giving the counts of digits in the first 
terms shows that the decimal digits are very uniformly distributed up to at least 
.
 |
OEIS |
10 |
100 |
 |
 |
 |
 |
 |
 |
 |
 |
| 0 |
A000000 |
0 |
10 |
108 |
952 |
9959 |
99814 |
999897 |
10002237 |
100010228 |
999996989 |
| 1 |
A000000 |
2 |
7 |
98 |
1005 |
10106 |
98924 |
1000114 |
10000179 |
99998381 |
1000042849 |
| 2 |
A000000 |
2 |
8 |
109 |
1004 |
9876 |
100436 |
1000208 |
9998091 |
99995645 |
999987069 |
| 3 |
A000000 |
2 |
11 |
82 |
980 |
10058 |
100191 |
999674 |
10004178 |
99995415 |
999984900 |
| 4 |
A000000 |
2 |
9 |
100 |
1016 |
10100 |
100024 |
1000126 |
10000054 |
100012725 |
1000008724 |
| 5 |
A000000 |
1 |
7 |
104 |
1001 |
10002 |
100155 |
999358 |
9998344 |
100002636 |
999970045 |
| 6 |
A000000 |
1 |
10 |
90 |
1032 |
9939 |
99886 |
1001246 |
10001665 |
100012683 |
1000007824 |
| 7 |
A000000 |
0 |
18 |
104 |
964 |
10008 |
100008 |
999359 |
9998646 |
99980315 |
999986743 |
| 8 |
A000000 |
0 |
12 |
113 |
1027 |
10007 |
100441 |
999452 |
9996550 |
99995120 |
1000025363 |
| 9 |
A000000 |
0 |
8 |
92 |
1019 |
9945 |
100121 |
1000566 |
10000056 |
99996852 |
999989494 |
REFERENCES:
Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Square Roots of Integers 2 to 15 in Various Bases 2 to 10: 88062 Binary Digits or Equivalent." Math. Comput. 23, 679, 1969.
Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Statistical Study of Digits of Some Square Roots of Integers in Various Bases." Math. Comput. 24, 455-473, 1970a.
Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "The Generalized Serial Test Applied to Expansions of Some Irrational Square Roots in Various Bases." Math. Comput. 24, 745-747, 1970b.
Sloane, N. J. A. Sequences A000129/M1314, A115377, A224871, and A229199 in "The On-Line Encyclopedia of Integer Sequences."
Yee, A. J. "y-cruncher - A Multi-Threaded Pi-Program." http://www.numberworld.org/y-cruncher/#Records.
