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Gauß's constant
From OeisWiki
√ 2 

,1 M (1, √ 2)
M (a, b) 
a 
b 

a 0 = a, b0 = b, an =
, bn =an − 1 + bn − 1 2 √ an − 1 bn − 1, n ≥ 1.
Carl Friedrich Gauß discovered (May 30, 1799) the following definite integral for this number

=1 M (1, √ 2)2 π ∫ 1 0
d x.1 √ 1 − x 4
Contents
Decimal expansion of Gauß’s constant
The decimal expansion of Gauß’s constant is

= 0.8346268416740731862814297327990468...1 M (1, √ 2)
giving the sequence of decimal digits (A014549)
 {8, 3, 4, 6, 2, 6, 8, 4, 1, 6, 7, 4, 0, 7, 3, 1, 8, 6, 2, 8, 1, 4, 2, 9, 7, 3, 2, 7, 9, 9, 0, 4, 6, 8, 0, 8, 9, 9, 3, 9, 9, 3, 0, 1, 3, 4, 9, 0, 3, 4, 7, 0, 0, 2, 4, 4, 9, 8, ...}
Continued fraction for Gauß’s constant
The simple continued fraction for Gauß’s constant is

giving the sequence of integer part and partial quotients (A053002)
 {0, 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, 5, 2, 1, 1, 2, 2, 6, 9, 1, 1, 1, 3, 1, 2, 6, 1, 5, 1, 1, 2, 1, 13, 2, 2, 5, 1, 2, 2, 1, 5, 1, 3, 1, 3, 1, 2, 2, ...}
Reciprocal of Gauß’s constant
Decimal expansion of reciprocal of Gauß’s constant
The decimal expansion of reciprocal of Gauß’s constant is

M (1, √ 2) = 1.19814023473559220743992249228...
giving the sequence of decimal digits (A053004)
 {1, 1, 9, 8, 1, 4, 0, 2, 3, 4, 7, 3, 5, 5, 9, 2, 2, 0, 7, 4, 3, 9, 9, 2, 2, 4, 9, 2, 2, 8, 0, 3, 2, 3, 8, 7, 8, 2, 2, 7, 2, 1, 2, 6, 6, 3, 2, 1, 5, 6, 5, 1, 5, 5, 8, 2, ...}
Continued fraction for reciprocal of Gauß’s constant
The simple continued fraction for reciprocal of Gauß’s constant is
M (1, √ 2 ) = 1 +

giving the sequence of integer part and partial quotients (A053003)
 {1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, 5, 2, 1, 1, 2, 2, 6, 9, 1, 1, 1, 3, 1, 2, 6, 1, 5, 1, 1, 2, 1, 13, 2, 2, 5, 1, 2, 2, 1, 5, 1, 3, 1, 3, 1, 2, 2, ...}