<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=content-type content=text/html;charset=us-ascii>
<META content="MSHTML 6.00.2600.0" name=GENERATOR></HEAD>
<BODY bottomMargin=0 leftMargin=3 topMargin=0 rightMargin=3>
<DIV>Seqfans,<BR> Would someone
wish to extend the following sequence-counting sequences?</DIV>
<DIV>For positive integer values of m, they are defined by:</DIV>
<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."</DIV>
<DIV> </DIV>
<DIV>For m=3: <BR>"Number of sequences (a_1, a_2,..., a_n) with a_1 = 1 <BR>such
that a_i < a_{i+1} <= 3*a_i for all i."<BR>I
get:<BR>1,2,10,114,2970,182402,27392682, </DIV>
<DIV> </DIV>
<DIV>An example of the tree being enumerated for m=3 is:
<BR>1<BR>| \<BR>2
3<BR>| \ \ \ \ \ \ \
\ \ <BR>3,4,5,6 4,5,6,7,8,9<BR>etc.
...<BR>-------------------------------------------------------<BR>For m=4:
<BR>"Number of sequences (a_1, a_2,..., a_n) with a_1 = 1 <BR>such that a_i <
a_{i+1} <= 4*a_i for all i."<BR>I get:<BR>1,3,27,693,52812,12628008,</DIV>
<DIV> </DIV>
<DIV>-------------------------------------------------------<BR>For m=5:
<BR>"Number of sequences (a_1, a_2,..., a_n) with a_1 = 1 <BR>such that a_i <
a_{i+1} <= 5*a_i for all i."<BR>I get:<BR>1,4,56,2704,481376,337587520,</DIV>
<DIV> </DIV>
<DIV>-------------------------------------------------------</DIV>
<DIV> </DIV>
<DIV>These sequences are obvious variants of A008934 (where m=2):</DIV>
<DIV> </DIV>
<DIV>"Number of tournament sequences: sequences (a_1, a_2, ..., a_n) with<BR>a_1
= 1 such that a_i < a_{i+1} <= 2*a_i for all i."<BR>which
begins:<BR>1,1,2,7,41,397,6377,171886,7892642,627340987,87635138366,<BR>21808110976027,9780286524758582,7981750158298108606,<BR>11950197013167283686587,33046443615914736611839942</DIV>
<DIV> </DIV>
<DIV>Reference (Cook and Kleber) for A008934:<BR><A
href="http://www.combinatorics.org/Volume_7/PDF/v7i1r44.pdf">http://www.combinatorics.org/Volume_7/PDF/v7i1r44.pdf</A></DIV>
<DIV>which supplies a formula for A008934. </DIV>
<DIV>See formula in the associated table (with A008934 as column
1):</DIV>
<DIV><A
href="http://www.research.att.com/projects/OEIS?Anum=A093729">http://www.research.att.com/projects/OEIS?Anum=A093729</A><BR>
</DIV>
<DIV>Can the formula be extended for m>2? </DIV>
<DIV>Finding more terms seems very time-consuming without the use
</DIV>
<DIV>of some derived formula.</DIV>
<DIV> </DIV>
<DIV>Thanks,<BR> Paul</DIV></BODY></HTML>