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<DIV><FONT face=Arial size=2>Dear seqfan,</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>if we denote S(n) concatenation of
numbers</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>How many increasing subsequence we can extract of
S(n) ?</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Examples S(n) = successive numbers</FONT></DIV>
<DIV><FONT face=Arial size=2>S(6) = 123456</FONT></DIV>
<DIV><FONT face=Arial size=2>increasing subsequence are :</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>1,2,3,4,5,6</FONT></DIV>
<DIV><FONT face=Arial size=2>1,2,3,4,56</FONT></DIV>
<DIV><FONT face=Arial size=2>1,2,3,456</FONT></DIV>
<DIV><FONT face=Arial size=2>1,2,34,56</FONT></DIV>
<DIV><FONT face=Arial size=2>1,2,3456</FONT></DIV>
<DIV><FONT face=Arial size=2>1,23,456</FONT></DIV>
<DIV><FONT face=Arial size=2>1,23456</FONT></DIV>
<DIV><FONT face=Arial size=2>12,34,56</FONT></DIV>
<DIV><FONT face=Arial size=2>12,3456</FONT></DIV>
<DIV><FONT face=Arial size=2>123,456</FONT></DIV>
<DIV><FONT face=Arial size=2>123456</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>for n=6 whe have 11 increasing subsequences
(a(6)=11)</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>and our sequence is
1,2,3,5,7,11,15,22,51,...</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>if S(n) is concatenation of prime
numbers</FONT></DIV>
<DIV><FONT face=Arial size=2>our sequence is 1,2,3,5,8,14,25</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>this sequence are not in OEIS</FONT></DIV>
<DIV><FONT face=Arial size=2>How many decreasing or increasing subsequence we
can extract of S(n)</FONT></DIV>
<DIV><FONT face=Arial size=2>If S(n) is succesive numbers</FONT></DIV>
<DIV>If S(n) is prime numbers</DIV>
<DIV>If S(n) is odd or even numbers ?</DIV>
<DIV><FONT face=Arial size=2>we can apply several sequence </FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Thanks</FONT></DIV>
<DIV> </DIV>
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