**:**
Introduction and an example.

**Masakazu
Naito, Toshiyuki Yamauchi, Daisuke Minematsu and Ryohei Miyadera**

**Kwansei
Gakuin **

**ABSTRACT
**

In
this article we study the Josephus problem in both direction. In this
variant of the Josephus problem two numbers are to be eliminated at
the same time, but two processes of elimination go for different
directions. Suppose that there are
-numbers
and every
-th
numbers are to be eliminated. We denote the number that remains by
.

At a glance this Josephus problem looks like a simple puzzle and
nothing more than a good example for computer programming, but the
sequence
presents interesting self-similarity of graph and the self-similarity
of the sequence when each term is divided by a certain number.

We
have presented the self-similarities of this Josephus problem in "The
Self-Similarity of the Josephus problem and its Variants, Visual
Mathematics Volume 11, No. 2, 2009", but we have not proved the
existence of the self-similarity of the sequence.

In this article
we prove it using the recursive relations when
.