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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
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 1 02:46 EST 2021. Contains 349426 sequences. (Running on oeis4.)