Talk:MRB constant

Feel free to post any research about the MRB constant in this discussion page or add it to the MRB constant page if you are certain of the facts. — Marvin Ray Burns 23:57, 8 May 2011 (UTC)

Iterated continued fraction for C_MRB

Cf. Talk:Convergents constant — See what happens if you use those convergents in a generalized continued fraction, then use the new convergents in another generalized continued fraction, then... ad infinitum! This is a work in progress.

Continued fraction for C_MRB

Continued Fraction for C_MRB=${\tfrac {0}{1}},{\tfrac {1}{5}},{\tfrac {3}{16}},{\tfrac {31}{165}},{\tfrac {34}{181}},{\tfrac {65}{346}},{\tfrac {294}{1565}},{\tfrac {359}{1911}},{\tfrac {653}{3476}},{\tfrac {1012}{5387}},{\tfrac {1665}{8863}},{\tfrac {15997}{85154}},{\tfrac {33659}{179171}},{\tfrac {421570}{2244069}},{\tfrac {876799}{4667309}},{\tfrac {15327153}{81588322}},{\tfrac {31531105}{167843953}},{\tfrac {78389363}{417276228}},{\tfrac {109920468}{585120181}},{\tfrac {188309831}{1002396409}},{\tfrac {3311187595}{17625859134}},\ldots \,$

Notice in the numerators and denominators that

0+1 != 3, 1+3 != 31; (2 false)

3+31 = 34, 31+34 = 65; (2 true)

34+65 != 294; (1 false)

65+294 = 359, 294+359 = 653, 359+653 = 1012; (3 true)

1012+ 1665 != 15997, 15997+33659 != 421570, 421570+15327153 != 31531105; (3 false)

31531105+78389363 = 109920468; (1 true)

109920468+188309831 != 3311187595; (? false)

...

Is there a pattern?

I think what appears as a pattern is only the partial quotients {3, 10, 1, 1, 4, ...} of the continued fraction itself!

3*1+0 = 3, 3*5+1 = 16; (3)

10*3+1 = 31, 10*16+5 = 165; (10)

1*31+3 = 34, 1*165+16 = 181; (1)

1*34+31 = 65, 1*181+165 = 346; (1)

4*65+34 = 294, 4*346+181 = 1565; (4)

— Daniel Forgues 19:51, 29 June 2011 (UTC)

Is this notable? It might just be a fact for all numbers; I haven't checked. Using the partial denominators as multiples of the (n-1) "partial denominator's numerator and denominator," and add the n-2 "partial denominator's numerator and denominator," we get partial denominator n. I mean the following:

${\frac {1*3}{5*3+1}}={\frac {3}{16}}$

${\frac {3*10+1}{16*10+5}}={\frac {31}{165}}$

${\frac {31*1+3}{165*1+16}}={\frac {34}{181}}$

${\frac {34*1+31}{181*1+165}}={\frac {65}{346}}$

${\frac {65*4+34}{346*4+181}}={\frac {294}{1565}}$

${\frac {294*1+65}{1565*1+346}}={\frac {359}{1911}}$

${\frac {359*1+294}{1911*1+1565}}={\frac {653}{3476}}$

${\frac {653*1+359}{3476*1+1911}}={\frac {1012}{5387}}$

${\frac {1012*1+653}{5387*1+3476}}={\frac {1665}{8863}}$

${\frac {1665*9+1012}{8863*9+5387}}={\frac {15997}{85154}}$

${\frac {15997*1+1665}{85154*1+8863}}={\frac {17662}{94017}}$

${\frac {17662*1+15997}{94017*1+85154}}={\frac {33659}{179171}}$

${\frac {33659*12+17662}{179171*12+94017}}={\frac {421570}{2244069}}$

${\frac {421570*2+33659}{2244069*2+179171}}={\frac {876799}{4667309}}$

${\frac {876799*17+421570}{4667309*17+2244069}}={\frac {15327153}{81588322}}$

${\frac {15327153*2+876799}{81588322*2+4667309}}{\text{==}}{\frac {31531105}{167843953}}$

${\frac {31531105*2+15327153}{167843953*2+81588322}}={\frac {78389363}{417276228}}$

${\frac {78389363*1+31531105}{417276228*1+167843953}}={\frac {109920468}{585120181}}$

—Marvin Ray Burns 01:08, 30 June 2011 (UTC)

This is just getting back the partial quotients of MRB constant#Continued fraction for CMRB...

{0, 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, 17, 1, 6, 4, 1, 3, 3, ...}

— Daniel Forgues 22:32, 30 June 2011 (UTC)

Update for the MRB constant article

Mathworld has added some sums and integrals for the MRB constant that might be useful, or proofs of which would also be great, on the OEIS wiki page: http://mathworld.wolfram.com/MRBConstant.html. Proofs in some form are found in my post at https://community.wolfram.com/groups/-/m/t/366628?p_p_auth=SlzqS8Nx. 06:02, 24 March 2020 (EDT)