A076187 -id:A076187 - cp's OEIS frontend

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

Benoit Cloitre, Nov 02 2002

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

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)

Cf. A076152, A076154, A076187.

Cf. A007400, A004200, A006466.

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

c=1.2656250596046447753906250000000000007... = A076187.

More terms from Ralf Stephan, May 17 2005

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

A076152 Let c = Sum_{k>=0} 1/2^(k!). Sequence gives values of terms not congruent to 5 in the continued fraction for c.

Original entry on oeis.org

1, 3, 1, 3, 4, 1, 3, 3, 1, 3, 1, 2, 1, 3, 3, 1, 4, 3, 1, 3, 1, 2, 1, 3, 4, 1, 3, 3, 1, 2, 1, 3, 1, 3, 3, 1, 4, 3, 1, 3, 1, 2, 1, 3, 4, 1, 3, 3, 1, 3, 1, 2, 1, 3, 3, 1, 4, 3, 1, 2, 1, 3, 1, 3, 4, 1, 3, 3, 1, 2, 1, 3, 1, 3, 3, 1, 4, 3, 1, 3, 1, 2, 1, 3, 4, 1, 3, 3, 1, 3, 1, 2, 1, 3, 3, 1, 4, 3, 1, 3, 1, 2, 1, 3, 4

Offset: 1

Benoit Cloitre, Nov 02 2002

nonn

Appears to contain only 1,2,3 or 4; seems to be a pseudo-periodic sequence.

			The continued fraction for c is shown in A076157. "Big terms" are all congruent to 5.

Cf. A076154, A076157, A076187.

A076154 Let c = Sum_{k>=0} 1/2^(k!), sequence gives values of terms congruent to 5 of the continued fraction for c.

Original entry on oeis.org

4095, 4722366482869645213695, 4095, 3121748550315992231381597229793166305748598142664971150859156959625371738819765620120306103063491971159826931121406622895447975679288285306290175, 4095, 4722366482869645213695, 4095

Offset: 1

Benoit Cloitre, Nov 02 2002

nonn

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) ...

			The continued fraction for c is shown in A076157. The "big terms" are all congruent to 5.

Cf. A076152, A076157, A076187.

It seems that for n>=1, a(2n-1)=4095; a(4n-2)=4722366482869645213695 etc.

A363290 Decimal expansion of the unique x > 0 such that Sum_{n>=0} 1/x^(n!) = 1.

Original entry on oeis.org

2, 4, 2, 4, 4, 1, 0, 4, 4, 9, 0, 1, 5, 6, 5, 3, 2, 3, 6, 3, 7, 2, 3, 7, 4, 5, 9, 7, 0, 7, 9, 4, 9, 7, 0, 8, 4, 1, 9, 5, 8, 4, 7, 7, 3, 2, 7, 1, 4, 7, 6, 9, 4, 3, 4, 2, 1, 2, 6, 5, 5, 9, 0, 0, 1, 5, 2, 7, 9, 8, 7, 0, 6, 7, 0, 7, 5, 4, 7, 4, 6, 7, 5, 0, 9, 1, 3, 6, 4, 4, 0, 0, 7, 7, 0, 3, 5

Offset: 1

M. F. Hasler, May 26 2023

			For x = 2.4244... and remembering that 0! = 1, we have Sum_{n >= 0} 1/x^n! ~ 2 * 1/2.4244 + 1/2.4244^2 + 1/2.4244^6 + 1/2.4244^24 = 1.0000...

Cf. A076187.

default(realprecision, 100); solve(x=2,3, sum(n=0,19, x^-n!)-1)

x = 2.424410449015653236372374597079497084...

cp's OEIS Frontend

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

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A076152 Let c = Sum_{k>=0} 1/2^(k!). Sequence gives values of terms not congruent to 5 in the continued fraction for c.

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A076154 Let c = Sum_{k>=0} 1/2^(k!), sequence gives values of terms congruent to 5 of the continued fraction for c.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A363290 Decimal expansion of the unique x > 0 such that Sum_{n>=0} 1/x^(n!) = 1.

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Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula