[seqfan] Re: confused about toothpick sequence A139250!
Maximilian Hasler
maximilian.hasler at gmail.com
Thu Apr 16 08:13:15 CEST 2009
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
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