A319464 Simple continued fraction expansion of a constant t with partial denominators a(n) such that the continued fraction with fractional partial denominators (2*a(n) + 3)/3 yields the same value t.

2, 1, 7, 5, 1, 1, 2, 2, 2, 2, 9, 1, 1, 1, 1, 3, 2, 2, 9, 11, 4, 51, 4, 1, 4, 2, 1, 2, 3, 6, 3, 2, 2, 5, 3, 1, 1, 1, 17, 1, 44, 2, 2, 15, 2, 2, 2, 30, 1, 1, 16, 1, 1, 2, 6, 1, 1, 1, 1, 3, 2, 740, 1, 2, 6, 24, 1, 1, 6, 1, 18, 1, 2, 13, 1, 19, 1, 9, 3, 1, 1, 4, 1, 6, 1, 1, 2, 7, 2, 6, 6, 1, 4, 1, 4, 3, 6, 4, 1, 2, 1, 1, 3, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 6, 2, 1, 1, 1, 4, 5, 5, 2, 3, 5, 5, 1, 1, 2, 2, 1, 1, 2, 4, 19, 4, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 8, 1, 1, 5, 3

Offset

0,1

Comments

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

The largest real number 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 = 0..1100

Example

Constant t = 2.877736075299027930414865011045376567...
The constant equals the continued fraction with partial denominators a(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/(a(n) + ...))))))))))))))))))).
the constant also equals the continued fraction with fractional partial denominators (2*a(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*a(n) + 3)/3 + ...)))))))))))))))))).
The simple continued fraction expansion begins:
t = [2; 1, 7, 5, 1, 1, 2, 2, 2, 2, 9, 1, 1, 1, 1, 3, 2, 2, 9, 11, 4,
51, 4, 1, 4, 2, 1, 2, 3, 6, 3, 2, 2, 5, 3, 1, 1, 1, 17, 1, 44, 2,
2, 15, 2, 2, 2, 30, 1, 1, 16, 1, 1, 2, 6, 1, 1, 1, 1, 3, 2, 740, 1,
2, 6, 24, 1, 1, 6, 1, 18, 1, 2, 13, 1, 19, 1, 9, 3, 1, 1, 4, 1, 6,
1, 1, 2, 7, 2, 6, 6, 1, 4, 1, 4, 3, 6, 4, 1, 2, 1, 1, 3, 2, 2,
1, 1, 1, 2, 1, 1, 1, 4, 6, 2, 1, 1, 1, 4, 5, 5, 2, 3, 5, 5, 1,
1, 2, 2, 1, 1, 2, 4, 19, 4, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 8, 1,
1, 5, 3, 5, 3, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 19, 1, 1,
5, 2, 3, 1, 7, 1, 7, 2, 1, 1, 2, 4, 3, 2, 2, 1, 4, 1, 1, 14, 1,
7, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 8, 3, 2, 1,
5, 1, 3, 2, 1, 2, 116, 5, 3, 2, 1, 1, 1, 1, 1, 1, 3, 7, 1, 56, 1,
1, 1, 16, 6, 3, 1, 7, 3, 11, 2, 4, 1, 4, 2, 1, 2, 1, 2, 1, 1, 2,
1, 1, 1, 2, 8, 1, 6, 1, 2, 2, 25, 2, 5, 4, 2, 1, 1, 6, 16, 11, 7,
6, 3, 11, 1, 1, 7, 1, 11, 1, 1, 12, 2, 1, 13, 2, 2, 13, 2, 1, 1, 1,
1, 1, 1, 1, 3, 1, 1, 1, 1, 98, 14, 2, 1, 1, 1, 2, 2, 5, 1, 3, 8,
26, 1, 3, 1, 9, 1, 2, 4, 1, 6, 1, 2, 6, 2, 1, 7, 1, 2, 1, 5, 105,
1, 3, 58, 1, 1, 1, 144, 1, 499, 1, 2, 2, 1, 3, 2, 14, 1, 3, 1, 5, 1,
3, 1, 18, 77, 9, 32, 1, 3, 7, 1, 10, 3, 3, 1, 1, 5, 8, 1, 3, 4, 1,
1, 1, 2, 1, 79, 3, 1, 1, 2, 1, 4, 8, 1, 22, 5, 1, 9, 1, 5, 5, 1,
1, 2, 1, 1, 6, 2, 6, 7, 3, 2, 6, 6, 1, 12, 5, 1, 7, 10, 1, 3, 2,
5, 2, 1, 9, 1, 1, 1, 1, 1, 2, 2, 5, 1, 8, 3, 2, 1, 2, 1, 2, 3,
5, 6, 16, 1, 1, 1, 8, 5, 21, 3, 2, 5, 1, 48, 1, 13, 2, 7, 1, 1, 4,
44, 1, 1, 3, 1, 1, 1, 1, 1, 1, 22, 1, 1, 2, 9, 3, 2, 3, 3, 3, 4,
4, 13, 1, 65, 1, 1, 7, 1, 1, 3, 1, 18, 7, 2, 1, 2, 3, 8, 4, 10, 1,
9, 1, 1, 4, 4, 2, 15, 2, 1, 1, 51, 14, 5, 2, 10, 1, 2, 28, 4, 9, 1,
1, 3, 1, 1, 1, 5, 1, 2, 1, 82, 9, 3, 1, 1, 2, 212, 8, 1, 1, 1, 100,
1, 3, 1, 4, 2, 1, 1, 3, 19, 2, 1, 1, 1, 5, 1, 10, 7, 2, 1, 1, 5,
2, 5, 9, 1, 3, 9, 1, 1, 40, 74, 3, 1, 1, 1, 1, 11, 1, 1, 1, 7, 1,
48, 1, 2, 1, 1, 1, 3, 2, 4, 4, 1, 2, 3, 1, 1, 1, 19, 7, 54, 1, 2,
3, 3, 1, 4, 170, 6, 3, 3, 11, 2, 1, 1, 35, 2, 2, 1, 5, 1, 2, 1, 4,
1, 1, 2, 1, 3, 4, 2, 1, 13, 12, 2, 4, 13, 4, 1, 2, 1, 1, 2, 1, 8,
6, 1, 12, 2, 2, 12, 2, 1, 411, 2, 3, 5, 3, 1, 1, 1, 1, 1, 2, 11, 6,
1, 2, 1, 1, 18, 3, 1, 20, 3, 1, 4, 4, 1, 3, 1, 10, 3, 2, 1, 16, 1,
1, 7, 4, 7, 2, 1, 11, 4, 17, 3, 36, 1, 2, 12, 4, 1, 2, 2, 1, 1, 1,
8, 1, 1, 82, 5, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 26, 1,
1, 8, 1, 19, 1, 4, 1, 2, 64, 1, 60, 1, 43, 1, 1, 8, 3, 1, 1, 41, 2,
4, 1, 30, 1, 8, 1, 4, 1, 5, 11, 1, 1, 1, 5, 1, 6, 3, 1, 1, 62, 2,
4, 1, 1, 2, 2, 17, 1, 5, 1, 2, 1, 3, 8, 8, 1, 6, 30, 1, 2, 5, 1,
3, 1, 8, 1, 9, 1, 9, 1, 1, 1, 2, 1, 3, 1, 160, 4, 4, 9, 1, 273, 1,
1, 4, 4, 1, 1, 5, 3, 7, 9, 1, 1, 6, 2, 1, 5, 19, 9, 1, 5, 5, 1,
2, 1, 123, 1, 32, 2, 2, 2, 4, 6, 26, 3, 2, 269, 1, 9, 11, 1, 1, 5, 1,
1, 1, 3, 5, 8, 4, 1, 7, 1, 1, 2, 1, 1, 1, 80, 1, 5, 2, 6, 2, 2,
2, 1, 16, 1, 11, 1, 3, 7, 2, 1, 1, 1, 3, 5, 1, 24, 1, 1, 6, 1, 1,
2, 2, 2, 1, 1, 2, 2, 4, 14, 2, 2, 2, 5, 1, 6, 2, 1, 1, 1, 30, 8,
2, 2, 4, 2, 2, 3, 2, 6, 1, 1, 1, 7, 1, 1, 2, 1, 1, 7, 2, 6, 4,
1, 1, 4, 1, 4, 2, 1, 4, 1, 11, 2, 1, 9, 2, 1, 1, 1, 1, 4, 4, 1,
1, 12, 12, 2, 7, 1, 3, 5, 3, 14, 4, 3, 1, 1, 1, 2, 2, 2, 4, 10, 3,
14, 13, 2, 1, 2, 3, 1, 1, 1, 1, 7, 2, 2, 2, 1, 1, 88, 10, 2, 1, 5,
11, 3, 31, 3, 1, 1, 1, 1, 1, 1, 1, 18, 3, 1, 2, 5, 1, 1, 1, 3, 2,
1, 1, 1, 5, 11, 1, 15, 2, 1, 1, 3, 6, 1, 1, 4, 1, 1, 29, 2, 2, 2,
3, 3, 1, 1, 33, 10, 1, 1, 1, 2, 2, 1, 16, 2, 4, 34, 3, 2, 2, 3, 2,
1, 1, 81, 4, 2, 3, 23, 2, 22, 3, 1, 2, 129, 1, 1, 2, 3, 4, 1, 1, 2,
1, 42, 1, 1, 3, 8, 1, 4, 1, 25, 1, 1, 6, 1, 6, 2, 4, 7, 1, 3, 6,
1, 25, 1, 15, 1, 1, 1, 2, 19, 1, 13, 2, 3, 23, 1, 1, 5, 1, 1, 11, 14,
1, 1, 5, 1, 1, 1, 10, 2, 6, 2, 14, 1, 11, 24, 5, 2, 2, 3, 1, 9, 4,
1, 2, 1, 1, 7, 3, 1, 4, 2, 1, 1, 18, 3, 1, 1, 1, 5, 1, 5, 4, 7, ...].

Crossrefs

Cf. A319463 (decimal expansion).

Sequence in context: A204771 A141512 A143877 * A019642 A339000 A248811

Adjacent sequences: A319461 A319462 A319463 * A319465 A319466 A319467

Keyword

nonn,cofr

Author

Paul D. Hanna, Sep 20 2018

Status

approved