A076157 Continued fraction expansion for c=sum_{k>=0} 1/2^(k!).

1, 3, 1, 3, 4, 4095, 1, 3, 3, 1, 3, 4722366482869645213695, 1, 2, 1, 3, 3, 1, 4095, 4, 3, 1, 3, 3121748550315992231381597229793166305748598142664971150859156959625371738819765620120306103063491971159826931121406622895447975679288285306290175

Offset: 1

Author

Benoit Cloitre, Nov 02 2002

Keywords

• cofr
• nonn

Comments

Observation: if b(k) denotes the sequence of all elements of the continued fraction for c, b(k) = 4095 if k==6 or 19 (mod 24); b(k) = 4722366482869645213695 if k==12 or 37 (mod 48); .... If b(k) is not congruent to 5 (mod 10), it seems that b(k) = 1,2,3 or 4 only.

Conjecture: a(3*2^n) = -1 + 2^[(n+1)((n+2)!) ]. - Ralf Stephan, May 17 2005

The conjecture follows from the theorem in Shallit's paper. The continued fraction has a "folded" overall structure. - Georg Fischer, Aug 29 2022

Links

• Rick L. Shepherd, Table of n, a(n) for n = 1..47
• Jeffrey O. Shallit, Simple Continued Fractions for Some Irrational Numbers II, J. Number Theory 14 (1982), 228-231.
• Rick L. Shepherd, Table of n, a(n) for n = 1..383 (has larger terms than b-files support)

Crossrefs

Cf. A076152, A076154, A076187.

Cf. A007400, A004200, A006466.

Programs

PARI

{allocatemem(220000000);
default(realprecision, 1000000);
contfrac(suminf(k=0, 1/(2^(k!))))}

Formula

c=1.2656250596046447753906250000000000007... = A076187.

Extensions

More terms from Ralf Stephan, May 17 2005

b-file, a-file, PARI program, and corrected conjecture by Rick L. Shepherd, Jun 07 2013