Roger Bagula's sequence A119565 (was Re: f(x) = ( f([x/2]) + f([x/3]) ) / 2)

[Dean Hickerson]

I wrote:

> Note that A119565 doesn't actually satisfy the equation by which it's
> supposed to be defined:
>
> %S A119565 1,2,3,5,7,10,12,14,17,20,23,27,31,36,42,49,57,66,76,88,101
> %N A119565 a(n) = Floor[a[n - 1] + 1 + (a[n - 2] + a[n - 3 + a[n - 4]])/6]
>
> From the Mathematica program it's clear that the values were computed by
> first initializing all values of a(n) to 0, setting a(0)=1, ..., a(3)=5,
> and then replacing a(n) by
>
>     floor(a(n-1) + 1 + (a(n-2) + a(n-3 + a(n-4)))/6)
>
> for n=4, 5, 6, ...
...
> So the computed sequence actually satisfies the simpler recurrence
>
>     a(n) = floor(a(n-1) + 1 + a(n-2)/6)  for n>=6.                    (*)
>
> If the sequence is to be kept in the OEIS then someone should fix the
> definition and replace the Mathematica code by something simpler, like:
>
>     a[0]=1;a[1]=2;a[2]=3;a[3]=5;a[4]=7;a[5]=10;
>
>     a[n_]:=a[n]=Floor[a[n-1]+1+a[n-2]/6];

I asked Mr. Bagula whether he intended to submit the sequence whose terms
he gave, and which satisfies (*), or one of the (perhaps infinitely many)
sequences that satisfy

    a(n) = floor(a(n-1) + 1 + (a(n-2) + a(n-3 + a(n-4)))/6).          (**)

I've received several messages from Mr. Bagula, which make it clear
that he wanted the sequence whose terms were given, not one which
satisfies (**).


The sequence entry also says:

> %C A119565 A chaotic recursive sequence with prime like behavior.

I asked:

> What's prime-like about it?

I haven't received an answer to that.


Mr. Bagula would prefer that I not edit his sequence, and I've promised
not to.  Is someone else willing to do the editing?

Dean Hickerson
dean at math.ucdavis.edu