Can All Coefficients in a Continued Fraction Be the Same Integer?

[Klaus_Hoffmann]

Hi, my question is given the recurrence relation for the convergents, could we construct a continued fraction so..

$$\alpha = a_{0}+ \frac{b_{0}}{a_{1}+\frac{b_{1}}{a_{2}}+...$$

all the coefficients a's and b's are equal to a certain integer ?

for example if all the coefficients (numerators and denomiators)

* are one we have just the Fibonacci (Golden ratio) constant $$\frac{2}{\sqrt 5 -1}$$

* are two we have exactly $$\sqrt 2 +1$$

i the sense that expanding the 2 numbers above their continued fraction is made only by 1 or 2, but can we construct a general continued fraction with all the numbers equal to 3,4,5,6,...

 

[ramsey2879]

Hi, my question is given the recurrence relation for the convergents, could we construct a continued fraction so..

$$\alpha = a_{0}+ \frac{b_{0}}{a_{1}+\frac{b_{1}}{a_{2}}+...$$

all the coefficients a's and b's are equal to a certain integer ?

for example if all the coefficients (numerators and denomiators)

* are one we have just the Fibonacci (Golden ratio) constant $$\frac{2}{\sqrt 5 -1}$$

* are two we have exactly $$\sqrt 2 +1$$

i the sense that expanding the 2 numbers above their continued fraction is made only by 1 or 2, but can we construct a general continued fraction with all the numbers equal to 3,4,5,6,...

Something doesn't seem right. If making all the a's and b's 1 gives the Fibonacci (golden ratio) constant, wouldn't making all the a's and b's 2 simply be 2 times the Golden ration rather than $$\sqrt 2 + 1$$.

Postscript. On the other hand: It seemed to me that any similar manner of construction a continued fraction will give an irrational number of some fixed value which would be a multiple of the continued fraction comprising only ones, but then n/n = 1 so I guess I made a mistake. Anyway at least the numbers are completely predefined.