A319464 - OEIS

%I #20 Sep 30 2018 06:51:13

%S 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,

%T 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,

%U 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

%N 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.

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

%C The largest real number with the same property equals 3 + 1/t = 3.3474953831...

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

%C Is this constant transcendental?

%H Paul D. Hanna, <a href="/A319464/b319464.txt">Table of n, a(n) for n = 0..1100</a>

%e Constant t = 2.877736075299027930414865011045376567...

%e The constant equals the continued fraction with partial denominators a(n):

%e 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) + ...))))))))))))))))))).

%e the constant also equals the continued fraction with fractional partial denominators (2*a(n) + 3)/3:

%e 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 + ...)))))))))))))))))).

%e The simple continued fraction expansion begins:

%e t = [2; 1, 7, 5, 1, 1, 2, 2, 2, 2, 9, 1, 1, 1, 1, 3, 2, 2, 9, 11, 4,

%e 51, 4, 1, 4, 2, 1, 2, 3, 6, 3, 2, 2, 5, 3, 1, 1, 1, 17, 1, 44, 2,

%e 2, 15, 2, 2, 2, 30, 1, 1, 16, 1, 1, 2, 6, 1, 1, 1, 1, 3, 2, 740, 1,

%e 2, 6, 24, 1, 1, 6, 1, 18, 1, 2, 13, 1, 19, 1, 9, 3, 1, 1, 4, 1, 6,

%e 1, 1, 2, 7, 2, 6, 6, 1, 4, 1, 4, 3, 6, 4, 1, 2, 1, 1, 3, 2, 2,

%e 1, 1, 1, 2, 1, 1, 1, 4, 6, 2, 1, 1, 1, 4, 5, 5, 2, 3, 5, 5, 1,

%e 1, 2, 2, 1, 1, 2, 4, 19, 4, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 8, 1,

%e 1, 5, 3, 5, 3, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 19, 1, 1,

%e 5, 2, 3, 1, 7, 1, 7, 2, 1, 1, 2, 4, 3, 2, 2, 1, 4, 1, 1, 14, 1,

%e 7, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 8, 3, 2, 1,

%e 5, 1, 3, 2, 1, 2, 116, 5, 3, 2, 1, 1, 1, 1, 1, 1, 3, 7, 1, 56, 1,

%e 1, 1, 16, 6, 3, 1, 7, 3, 11, 2, 4, 1, 4, 2, 1, 2, 1, 2, 1, 1, 2,

%e 1, 1, 1, 2, 8, 1, 6, 1, 2, 2, 25, 2, 5, 4, 2, 1, 1, 6, 16, 11, 7,

%e 6, 3, 11, 1, 1, 7, 1, 11, 1, 1, 12, 2, 1, 13, 2, 2, 13, 2, 1, 1, 1,

%e 1, 1, 1, 1, 3, 1, 1, 1, 1, 98, 14, 2, 1, 1, 1, 2, 2, 5, 1, 3, 8,

%e 26, 1, 3, 1, 9, 1, 2, 4, 1, 6, 1, 2, 6, 2, 1, 7, 1, 2, 1, 5, 105,

%e 1, 3, 58, 1, 1, 1, 144, 1, 499, 1, 2, 2, 1, 3, 2, 14, 1, 3, 1, 5, 1,

%e 3, 1, 18, 77, 9, 32, 1, 3, 7, 1, 10, 3, 3, 1, 1, 5, 8, 1, 3, 4, 1,

%e 1, 1, 2, 1, 79, 3, 1, 1, 2, 1, 4, 8, 1, 22, 5, 1, 9, 1, 5, 5, 1,

%e 1, 2, 1, 1, 6, 2, 6, 7, 3, 2, 6, 6, 1, 12, 5, 1, 7, 10, 1, 3, 2,

%e 5, 2, 1, 9, 1, 1, 1, 1, 1, 2, 2, 5, 1, 8, 3, 2, 1, 2, 1, 2, 3,

%e 5, 6, 16, 1, 1, 1, 8, 5, 21, 3, 2, 5, 1, 48, 1, 13, 2, 7, 1, 1, 4,

%e 44, 1, 1, 3, 1, 1, 1, 1, 1, 1, 22, 1, 1, 2, 9, 3, 2, 3, 3, 3, 4,

%e 4, 13, 1, 65, 1, 1, 7, 1, 1, 3, 1, 18, 7, 2, 1, 2, 3, 8, 4, 10, 1,

%e 9, 1, 1, 4, 4, 2, 15, 2, 1, 1, 51, 14, 5, 2, 10, 1, 2, 28, 4, 9, 1,

%e 1, 3, 1, 1, 1, 5, 1, 2, 1, 82, 9, 3, 1, 1, 2, 212, 8, 1, 1, 1, 100,

%e 1, 3, 1, 4, 2, 1, 1, 3, 19, 2, 1, 1, 1, 5, 1, 10, 7, 2, 1, 1, 5,

%e 2, 5, 9, 1, 3, 9, 1, 1, 40, 74, 3, 1, 1, 1, 1, 11, 1, 1, 1, 7, 1,

%e 48, 1, 2, 1, 1, 1, 3, 2, 4, 4, 1, 2, 3, 1, 1, 1, 19, 7, 54, 1, 2,

%e 3, 3, 1, 4, 170, 6, 3, 3, 11, 2, 1, 1, 35, 2, 2, 1, 5, 1, 2, 1, 4,

%e 1, 1, 2, 1, 3, 4, 2, 1, 13, 12, 2, 4, 13, 4, 1, 2, 1, 1, 2, 1, 8,

%e 6, 1, 12, 2, 2, 12, 2, 1, 411, 2, 3, 5, 3, 1, 1, 1, 1, 1, 2, 11, 6,

%e 1, 2, 1, 1, 18, 3, 1, 20, 3, 1, 4, 4, 1, 3, 1, 10, 3, 2, 1, 16, 1,

%e 1, 7, 4, 7, 2, 1, 11, 4, 17, 3, 36, 1, 2, 12, 4, 1, 2, 2, 1, 1, 1,

%e 8, 1, 1, 82, 5, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 26, 1,

%e 1, 8, 1, 19, 1, 4, 1, 2, 64, 1, 60, 1, 43, 1, 1, 8, 3, 1, 1, 41, 2,

%e 4, 1, 30, 1, 8, 1, 4, 1, 5, 11, 1, 1, 1, 5, 1, 6, 3, 1, 1, 62, 2,

%e 4, 1, 1, 2, 2, 17, 1, 5, 1, 2, 1, 3, 8, 8, 1, 6, 30, 1, 2, 5, 1,

%e 3, 1, 8, 1, 9, 1, 9, 1, 1, 1, 2, 1, 3, 1, 160, 4, 4, 9, 1, 273, 1,

%e 1, 4, 4, 1, 1, 5, 3, 7, 9, 1, 1, 6, 2, 1, 5, 19, 9, 1, 5, 5, 1,

%e 2, 1, 123, 1, 32, 2, 2, 2, 4, 6, 26, 3, 2, 269, 1, 9, 11, 1, 1, 5, 1,

%e 1, 1, 3, 5, 8, 4, 1, 7, 1, 1, 2, 1, 1, 1, 80, 1, 5, 2, 6, 2, 2,

%e 2, 1, 16, 1, 11, 1, 3, 7, 2, 1, 1, 1, 3, 5, 1, 24, 1, 1, 6, 1, 1,

%e 2, 2, 2, 1, 1, 2, 2, 4, 14, 2, 2, 2, 5, 1, 6, 2, 1, 1, 1, 30, 8,

%e 2, 2, 4, 2, 2, 3, 2, 6, 1, 1, 1, 7, 1, 1, 2, 1, 1, 7, 2, 6, 4,

%e 1, 1, 4, 1, 4, 2, 1, 4, 1, 11, 2, 1, 9, 2, 1, 1, 1, 1, 4, 4, 1,

%e 1, 12, 12, 2, 7, 1, 3, 5, 3, 14, 4, 3, 1, 1, 1, 2, 2, 2, 4, 10, 3,

%e 14, 13, 2, 1, 2, 3, 1, 1, 1, 1, 7, 2, 2, 2, 1, 1, 88, 10, 2, 1, 5,

%e 11, 3, 31, 3, 1, 1, 1, 1, 1, 1, 1, 18, 3, 1, 2, 5, 1, 1, 1, 3, 2,

%e 1, 1, 1, 5, 11, 1, 15, 2, 1, 1, 3, 6, 1, 1, 4, 1, 1, 29, 2, 2, 2,

%e 3, 3, 1, 1, 33, 10, 1, 1, 1, 2, 2, 1, 16, 2, 4, 34, 3, 2, 2, 3, 2,

%e 1, 1, 81, 4, 2, 3, 23, 2, 22, 3, 1, 2, 129, 1, 1, 2, 3, 4, 1, 1, 2,

%e 1, 42, 1, 1, 3, 8, 1, 4, 1, 25, 1, 1, 6, 1, 6, 2, 4, 7, 1, 3, 6,

%e 1, 25, 1, 15, 1, 1, 1, 2, 19, 1, 13, 2, 3, 23, 1, 1, 5, 1, 1, 11, 14,

%e 1, 1, 5, 1, 1, 1, 10, 2, 6, 2, 14, 1, 11, 24, 5, 2, 2, 3, 1, 9, 4,

%e 1, 2, 1, 1, 7, 3, 1, 4, 2, 1, 1, 18, 3, 1, 1, 1, 5, 1, 5, 4, 7, ...].

%Y Cf. A319463 (decimal expansion).

%K nonn,cofr

%O 0,1

%A _Paul D. Hanna_, Sep 20 2018