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<DIV>Seqfans, </DIV>
<DIV> Consider the nice sequence A006336: </DIV>
<DIV>a(n) = a(n-1) + a(n-1 -<STRONG> number of even terms so far</STRONG>).
</DIV>
<DIV><A
href="http://www.research.att.com/~njas/sequences/A006336">http://www.research.att.com/~njas/sequences/A006336</A></DIV>
<DIV>begins:
[1,2,3,5,8,11,16,21,29,40,51,67,88,109,138,167,207,258,309,376,...].</DIV>
<DIV> </DIV>
<DIV>My COMMENT (NOT submitted to OEIS): </DIV>
<DIV>-----------------------------------------------------------</DIV>
<DIV>It seems that A006336 can be generated by a rule using the golden
ratio:</DIV>
<DIV> </DIV>
<DIV>a(n) = a(n-1) + a([n/<STRONG>Phi</STRONG>]) for n>1 with
a(1)=1 where Phi = (sqrt(5)+1)/2, </DIV>
<DIV> </DIV>
<DIV>
<DIV>i.e., the number of even terms up to position n-1 equals: </DIV>
<DIV>n-1 - [n/<STRONG>Phi</STRONG>] for n>1 where Phi = (sqrt(5)+1)/2.
</DIV></DIV>
<DIV> </DIV>
<DIV>(PARI): </DIV>
<DIV>a(n) = if(n==1,1, a(n-1) + a( floor(n/((sqrt(5)+1)/2)) ) )</DIV>
<DIV>
<DIV>-----------------------------------------------------------</DIV></DIV>
<DIV> </DIV>
<DIV>Would someone verify if these are indeed equivalent definitions, at least
empirically? </DIV>
<DIV>Or, what is the first position in which terms are NOT equal?
</DIV>
<DIV> </DIV>
<DIV>If these are equivalent, then this is another unexpected appearance of
that ubiquitous constant. </DIV>
<DIV>Thanks, </DIV>
<DIV> Paul </DIV></BODY></HTML>