[seqfan] Re: A171791: Surprising Conjecture for Odd Terms

[Sean A. Irvine]

Hi,

I extended Paul's calculation of A171791 to term a(1028) and found no
counter-example to the property he has observed. In particular, a(683) and
a(1025) are odd and all intervening terms even.   I have submitted an
updated b-file for the sequence.

Regards,
Sean.


On Mon, Apr 21, 2014 at 3:08 PM, Paul D Hanna <pauldhanna at juno.com> wrote:

>  SeqFans,
>      Consider the conjecture stated in https://oeis.org/A171791 :
> "It appears that for k>0,
>      a(k) is odd iff k = 2*A003714(n) + 1  for n>=0,
> where A003714 is the fibbinary numbers
> (integers whose binary representation contains no consecutive ones);
> this is true for at least the first 520 terms."
>
> That is, the odd terms are located at positions:
> [0, 1, 3,  5,  9,11,  17,19,21,  33,35,37,41,43,  65,67,69,73,75,81,83,85,
>  129,131,133,137,139,145,147,149,161,163,165,169,171, ...].
>
> That fibbinary numbers would be involved is surprising since
> the definition of A171791 involves square powers of the g.f.:
> G.f. A(x) satisfies: [x^n] A(x)^((n+1)^2) = 0 for n>1 with a(0)=a(1)=1.
>
> Could someone give a rationale as to why such an unexpected property would
> hold?
>
> It would also be nice if the conjecture could be tested up to more terms
> than the initial 520.
>
> Thanks,
>       Paul
>
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