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).
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
about the second?
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.