A319463 Least constant t > 0 such that the simple continued fraction of t may be expressed by partial denominators A319464(n) as well as fractional partial denominators (2*A319464(n) + 3)/3.

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

Offset

1,1

Comments

This constant t is the least positive real number satisfying the definition.

Largest constant with the same property equals 3 + 1/t = 3.3474953831...

Note that (sqrt(13) + 3)/2 = [3; 3, 3, 3, ...] also satisfies the property.

Is this constant transcendental?

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1101

Example

Constant t = 2.877736075299027930414865011045376567...
The constant equals the continued fraction with partial denominators A319464(n):
t = 2 + 1/(1 + 1/(7 + 1/(5 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(2 + 1/(2 + 1/(9 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(3 + 1/(2 + 1/(2 + 1/(9 + 1/(11 + ... + 1/(A319464(n) + ...)))))))))))))))))))).
also equals the continued fraction with fractional partial denominators (2*A319464(n) + 3)/3:
t = 7/3 + 1/(5/3 + 1/(17/3 + 1/(13/3 + 1/(5/3 + 1/(5/3 + 1/(7/3 + 1/(7/3 + 1/(7/3 + 1/(7/3 + 1/(7 + 1/(5/3 + 1/(5/3 + 1/(5/3 + 1/(5/3 + 1/(3 + 1/(7/3 + 1/(7/3 + 1/(7 + 1/(25/3 + ... + 1/( (2*A319464(n) + 3)/3 + ...)))))))))))))))))))).

Crossrefs

Cf. A319464 (continued fraction).

Sequence in context: A213930 A394908 A393257 * A379711 A079031 A388227

Adjacent sequences: A319460 A319461 A319462 * A319464 A319465 A319466

Keyword

nonn,cons

Author

Paul D. Hanna, Sep 20 2018

Status

approved