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<DIV><FONT face=Courier size=2>Superseeker can sometimes be
miraculous:</FONT></DIV>
<DIV><FONT face=Courier size=2></FONT> </DIV>
<DIV><FONT face=Courier size=2>I fed it a sequence
1,2,6,20,76,296,1240,5200,22960,100512,458592</FONT></DIV>
<DIV><FONT face=Courier size=2></FONT> </DIV>
<DIV><FONT face=Courier size=2>generated by counting the length of the string
</FONT></DIV>
<DIV><FONT face=Courier size=2>{0} ; {-1,1} ; {-2,0,2, -2,0,2};<BR>or "replace k
with {-1-Abs[k], ... , 1+Abs[k]} with increment 2".</FONT></DIV>
<DIV><FONT face=Courier size=2></FONT> </DIV>
<DIV><FONT face=Courier size=2>and it replied with:</FONT></DIV>
<DIV><FONT face=Courier size=2>[-2n^2+n^3+(4n+5n^2) a[n] + (-2-6n-6n^2-2n^3)
a[n]^2, revogf]<BR></FONT></DIV>
<DIV><FONT face=Courier size=2>this juggles into</FONT></DIV>
<DIV><FONT face=Courier size=2><FONT
size=3>CoefficientList[InverseSeries[</FONT></FONT></DIV>
<DIV><FONT face=Courier size=2><FONT
size=3>Series[(-4n-5n^2+n^2Sqrt[1+8n+8n^2])/(2(-2-6n-6n^2-2n^3)),{n,0,12}]</FONT></FONT><FONT
face=Courier size=2><FONT size=3> ],n]</FONT><BR></FONT></DIV>
<DIV><FONT face=Courier size=2>How does <FONT size=3>LISTTOALGEQ</FONT> work
this? Who wrote that?</FONT></DIV>
<DIV><FONT face=Courier size=2></FONT> </DIV>
<DIV><FONT face=Courier size=2>same sequence, starting from {1}
gives</FONT></DIV>
<DIV><FONT face=Courier size=2><FONT size=3>{1} ; {-2,0,2} ; {-3,-1,1,3 , -1,1 ,
-3,-1,1,3}</FONT><BR>with lengths counted by</FONT></DIV>
<DIV><FONT face=Courier size=2><FONT size=3>(2, 6, 20, 76, 296, 1240, 5200,...)
/2</FONT></FONT></DIV>
<DIV><FONT face=Courier size=2><FONT size=3></FONT></FONT> </DIV>
<DIV><FONT face=Courier size=2><FONT size=3>if you count the sums of the Abs
instead of lengths,</FONT></DIV>
<DIV>you get, starting from {1}:<BR>1, 4, 18, 72, 324, 1360, 6280<BR>which is
just<BR>{3, 10, 38, 148, 620, 2600} - {2, 6, 20, 76, 296, 1240}<BR></DIV>
<DIV>and starting from {0}:</DIV>
<DIV>0, 2, 8, 36, 144, 648, 2720 </DIV>
<DIV>which is twice the previous shifted right.</DIV>
<DIV> </DIV>
<DIV>************************</DIV>
<DIV>Counting lengths of nested lists generates familiar seq's,</DIV>
<DIV>as </DIV>
<DIV>A000108 from k -> 0,..,k+1</DIV>
<DIV>A001003 from k-> 0,..,k+1,..,0</DIV>
<DIV>A006319 from k -> -Abs[k],..,Abs[k]+1 </DIV>
<DIV> with again a link to (inverse binomial transform of
<FONT face=Courier size=2>A071356).</FONT></DIV>
<DIV> </DIV>
<DIV>************************</DIV>
<DIV> </DIV>
<DIV>Wouter.</DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV> </DIV></FONT></BODY></HTML>