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<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'><!-- Converted from text/enriched format -->He
SeqFans,</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>In Neil Sloane’s “Classic
Sequences” <a
href="http://www.research.att.com/~njas/sequences/classic.html">http://www.research.att.com/~njas/sequences/classic.html</a>
</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>there are some “partial
complements” whose union is the whole complement. For example, on
page 1, the</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>Wythoff Array
appears. Look at column 3; its complement, C, is the ordered union of
columns 4,5,6,…</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>The sequence (2,3,5,8,13,…)
in row 1 is the “1<sup>st</sup> partial complement” of C;</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>the sequence
(7,11,18,29,47,…) in row 2 is the “2<sup>nd</sup> partial
complement” of C; and so on.</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>In “Classic
Sequences,” scroll down to “Other Links” and visit “Partial
Complements and Transposable</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>Dispersions” (that
takes you to JIS, where you can scroll down to Article 04.1.6) – or,</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>for the pdf, just click <a
href="http://www.math.uwaterloo.ca/JIS/VOL7/Kimberling/kimber67.pdf">http://www.math.uwaterloo.ca/JIS/VOL7/Kimberling/kimber67.pdf</a>
)</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>Section 1 of the article
defines “ith partial complement” for i = 1,2,3,… The
triangular numbers (and other</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>such sequences) are fixed
points under certain mappings. John and I kicked this ball downfield aways
</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>but finally (p21) tossed up
our hands and wrote “It is hoped that someone will prove those
conjectures.”</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>(See the Abstract for a
simple statement of the main conjecture.)</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'>Clark Kimberling</span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal style='text-indent:.5in'><font size=2 color=navy face=Arial><span
style='font-size:10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>_____________________________________________________</span></font></p>
<p class=MsoNormal style='margin-left:.5in'><font size=2 face=Tahoma><span
style='font-size:10.0pt;font-family:Tahoma'>-----Original Message-----<br>
<b><span style='font-weight:bold'>From:</span></b> benoit
[mailto:abcloitre@wanadoo.fr] <br>
<b><span style='font-weight:bold'>Sent:</span></b> </span></font><font size=2 face=Tahoma><span style='font-size:10.0pt;font-family:Tahoma'>Friday, March
26, 2004</span></font><font size=2 face=Tahoma><span style='font-size:10.0pt;
font-family:Tahoma'> </span></font><font size=2 face=Tahoma><span
style='font-size:10.0pt;font-family:Tahoma'>12:03 AM</span></font><font
size=2 face=Tahoma><span style='font-size:10.0pt;font-family:Tahoma'><br>
<b><span style='font-weight:bold'>To:</span></b> seqfan@ext.jussieu.fr<br>
<b><span style='font-weight:bold'>Subject:</span></b> Re: In One Sequence
Or Another (Non-Fibonacci)</span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'> </span></font></p>
<div>
<p class=MsoNormal style='margin-left:.5in'><font size=3 face=Arial><span
style='font-size:12.0pt;font-family:Arial'>There is also the complement of
triangular numbers with a simple formula and some properties : </span></font></p>
</div>
<p class=MsoNormal style='margin-left:.5in'><font size=3 face=Arial><span
style='font-size:12.0pt;font-family:Arial'> </span></font></p>
<div>
<p class=MsoNormal style='margin-left:.5in'><font size=3 face=Arial><span
style='font-size:12.0pt;font-family:Arial'>http://www.research.att.com/projects/OEIS?Anum=A014132
</span></font></p>
</div>
<p class=MsoNormal style='margin-left:.5in'><font size=3 face=Arial><span
style='font-size:12.0pt;font-family:Arial'> </span></font></p>
<div>
<p class=MsoNormal style='margin-left:.5in'><font size=3 face=Arial><span
style='font-size:12.0pt;font-family:Arial'>Benoit Cloitre </span></font></p>
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