## SMARANDACHE CONTINUED FRACTIONS

### Jose Castillo

Navajo Community College, Tsaile, Arizona, USA

1) A Smarandache Simple Continued Fraction is a fraction of the form:

 1 a(1)  +  ----------------------------------------------- 1 a(2) +  ------------------------------------- 1 a(3) + ---------------------------- 1 a(4) +  ---------------- a(5)  .  .  .

where a(n), for n >= 1, is a Smarandache type Sequence, Sub-Sequence, or Function.

2) And a Smarandache General Continued Fraction is a fraction of the form:

 b(1) a(1)  +  ----------------------------------------------- b(2) a(2) +  ------------------------------------- b(3) a(3) + ---------------------------- b(4) a(4) +  ---------------- a(5)  .  .  .

where a(n) and b(n), for n >= 1, are both Smarandache type Sequences,

Sub-Sequences, or Functions.

(Over 200 such sequences are listed in Sloane's database of Encyclopedia of

Integer sequences -- online).

URL: http://www.research.att.com/~njas/sequences/

For example:

a) If we consider the smarandache consecutive sequence:

1, 12, 123, 1234, 12345, ..., 123456789101112, ...

we form a smarandache simple continued fraction:

 1 1  +  ----------------------------------------------- 1 12 +  ------------------------------------- 1 123 + ---------------------------- 1 1234 +  ---------------- 12345  .  .  .

b) If we include the smarandache reverse sequence:

1, 21, 321, 4321, 54321, ..., 121110987654321, ...

to the previous one we get a smarandache general continued fraction:

 1 1  +  ----------------------------------------------- 21 12 +  ------------------------------------- 321 123 + ---------------------------- 4321 1234 +  ---------------- 12345  .  .  .

With a mathematics software program, it is possible to calculate such continued

fractions to see which ones of them converge, and eventually to make conjectures,

or to algebraically prove those converging towards certain constants.

Open Problem: Calculate each above continued fraction.

The previous example of continued fraction is convergent(Dodge[1]), but what

### References

[1] Castillo, Jose, "Smarandache Continued Fractions", Bulletin of Pure and

Applied Sciences, Delhi, India, Vol. 17E, No. 1, 149-151, 1998.

[2] Castillo, Jose, "Smarandache Continued Fractions", Smarandache Notions

Journal, Vol. 9, No. 1-2, 40-42, 1998.

[3] Dodge, Clayton W., Letter to the Author, August 4, 1998.

[4] Smarandoiu, Stefan, "Convergence of Smarandache Continued Fractions",

Abstracts of Papers Presented to the American Mathematical Society, Vol. 17,

No. 4, Issue 106, 680, 1996.