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<DIV>Seqfans,</DIV>
<DIV> Thanks, Brendan, for the
comments regarding the recurrence. </DIV>
<DIV>I have found 3 distinct recurrences, and I am working on
one more. </DIV>
<DIV>There are 2 surprising recurrences involving powers of matrices,
and </DIV>
<DIV>also I found the generalization of the Cook-Kleber
recurrence for all m>1.</DIV>
<DIV> </DIV>
<DIV>
<DIV>I am preparing this whole family of sequences for submission
to OEIS </DIV>
<DIV>(it may take me up to a week before they are finalized
and submitted).</DIV>
<DIV> </DIV>
<DIV>Question: since these sequences are a natural extension of A008934, </DIV>
<DIV>which is the number of tournament sequences (where m=2), </DIV>
<DIV>what name could I assign these related sequences? </DIV>
<DIV>Should I refer to them as "k-tournament sequences" - any suggestions?
</DIV>
<DIV> </DIV></DIV>
<DIV>I am referring to the sequence-counting sequences defined
by: </DIV>
<DIV>
<DIV>"Number of sequences (a_1, a_2,..., a_n) with a_1 = 1 <BR>such that a_i
< a_{i+1} <= m*a_i for all i." where m>1.</DIV></DIV>
<DIV> </DIV>
<DIV>For example, when m=3 we obtain these terms using the recurrences:</DIV>
<DIV>1,2,10,114,2970,182402,27392682,10390564242,10210795262650,<BR>26494519967902114,184142934938620227530,<BR>3466516611360924222460082,178346559667060145108789818842,<BR>25264074391478558474014952210052802
</DIV>
<DIV>which exactly agrees with explicit counting (that
obtained only the initial 8 terms).</DIV>
<DIV> </DIV>
<DIV>
<DIV>Thanks, </DIV></DIV>
<DIV> Paul</DIV></BODY></HTML>