[seqfan] Re: confused about toothpick sequence A139250!

[Maximilian Hasler]

oops... some more typos...

> since d(n)=a(n+1)-a(n) verifies
...
>  d( 2^k + m ) = d( 2^(k-1) + m )   for  2^(k-1) <= m < 3*2^(k-2)-2
>  d( 2^k + m ) = d( 2^(k-1) + m ) + 4   for  m = 3*2^(k-2)-2

should read (I think...)

 d( 2^k + m ) = d( 2^(k-2) + m )   for  2^(k-1) <= m < 3*2^(k-2)-2
 d( 2^k + m ) = d( 2^(k-2) + m ) + 4 = d( 2^k - 2 ) + 4   for  m = 3*2^(k-2)-2

and
> d( 2^k + m ) = d( 2^k - 1 ) + 24  for m = 3*2^(k-2)-1

could be simplified to
d( 2^k + m ) = 2^k + 24  for m = 3*2^(k-2)-1

Benoît Jubin asked for  lim inf  and  lim sup  of  a(n)/n².
I think that the lim sup equals the 2/3 obtained for n=2^k.
(For n=2^k-1, the largest value for all n>2^(k-1) is reached,
but it eventually tends to 2/3.)
The lim inf however seems to be <= 1/2.
Maybe the precise value could be found by using an interpolating
function a(x) whose derivative d(x) = a'(x) verifies
d( x ) = d( x - 2^(k-1) )  for  2^k < x < 2^k+2^(k-2)-1.
I think it is in (the middle of the 2nd half of) that range, where one
will find the values which produce the lim inf.

Maximilian