Journal of Integer Sequences, Vol. 11 (2008), Article 08.3.6 |

Department of Mathematics

University of Toronto

Toronto, Ontario M5S 2E4

Canada

**Abstract:**

We explore the effect of different values of the shift parameter
on the behavior of the family of meta-Fibonacci sequences defined by
the -term recursion

with the initial conditions for . We show that for any odd and non-negative integer the values in the sequence and are essentially the same. The only differences in these sequences are that each power of occurs precisely times in and times in . For even the frequency of in depends upon . We conjecture that for even the effect of the shift parameter is analogous to that for odd, in the sense that the only differences in the sequences and occur in the frequencies of the powers of ; specifically, each power of appears to occur precisely more times in than it does in .

with the initial conditions for . We show that for any odd and non-negative integer the values in the sequence and are essentially the same. The only differences in these sequences are that each power of occurs precisely times in and times in . For even the frequency of in depends upon . We conjecture that for even the effect of the shift parameter is analogous to that for odd, in the sense that the only differences in the sequences and occur in the frequencies of the powers of ; specifically, each power of appears to occur precisely more times in than it does in .

Received June 2 2008;
revised version received August 15 2008.
Published in *Journal of Integer Sequences*, August 17 2008.

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