Ternary analogue of A094913?

[Maximilian Hasler]

Some explicit formulae concerning A094913, A006697 and related seq.,
based on the conjecture that

A094913(n,base=2)=sum(k=1,n,min(base^k,n-k+1))
resp.
A006697(n,b=2)=sum(k=0,n,min(b^k,n-k+1))

This gives:

A006697(n,b) = (b^(m+1)-1)/(b-1) + (n-m)(n-m+1)/2

where m is the integer part of the solution to
   b^m = n+1-m
given by
  m = [ n+1-LambertW( b^(n+1) * log(b) ) / log(b) ]

and, obviously, A094913(n,b)=A006697(n,b)-1.

at least for A006697(n) the values obtained by my explicit formula
agree with all those listed there (n<=53).

(PARI) :
LambertW(y) = solve( X=1,log(y), X*exp(X)-y)
A006697(n,b=2) = local( m=floor(n+1-LambertW(b^(n+1)*log(b))/log(b)));
(b^(m+1)-1)/(b-1)+(n-m)*(n-m+1)/2


On Dec 14, 2007 9:07 AM, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> of course, cf an earlier reply of myself,
> and I don't have anything against the binary (or 2-color) version,
> but if we go on to base b=3 (or b=3 colours) then we have to do it for
> any number of colours b>2.
> However, I think this is already done, if the formula I gave earlier
> (Dec 11, 2007 8:48 AM) is correct:
>
> A094913(n,base=2)=sum(k=1,n,min(base^k,n-k+1))
>
> Splitting up the sum in 2 parts to get rid of the min() allows easily
> to get an explicit analytic expression for the n-the term of the
> sequence for any 'base' (= number of colours).
>
> Maximilian
>
>
>
> On Dec 14, 2007 8:49 AM, David Wilson <davidwwilson at comcast.net> wrote:
> > The sequence in question is not about binary number, it is a sequence about
> > strings over a 2-character alphabet encoded as binary numbers, and does not
> > concern the numerical values of the elements.  It is more akin to "necklaces
> > having beads of 2 colors".
> >
> >
> > ----- Original Message -----
> > From: "Maximilian Hasler" <maximilian.hasler at gmail.com>
> > To: "Jonathan Post" <jvospost3 at gmail.com>
> > Cc: "seqfan" <seqfan at ext.jussieu.fr>; "jonathan post" <jvospost2 at yahoo.com>
> > Sent: Tuesday, December 11, 2007 7:39 AM
> > Subject: Re: Ternary analogue of A094913?
> >
> >
> > > If you go on to base 3, then the door is open to do it in any base...
> > > :-( !
> >
> >
>
>
>
> --
> Maximilian F. Hasler (Maximilian.Hasler(AT)gmail.com)
>



-- 
Maximilian F. Hasler (Maximilian.Hasler(AT)gmail.com)