a puzzle sequence

[Paul C. Leopardi]

Hi all, 
There are some patterns here. Unfortunately not enough to lead me to a 
definitive answer. See details below.
Best regards, Paul Leopardi

On Tue, 26 Jul 2005 10:49 pm, N. J. A. Sloane wrote:
> SeqFans,  this just came in. Can anyone solve it?
> I don't know the answer.  NJAS
>
> > need help with a math thing
> >
> >
> > 87   711   43   269   65   555   39   421   17   ___    ___    ___
> >
> > What are three numbers?  What is the pattern?  What do they have in
> > common?
> >
> > If you could help it would be great!!

Some patterns:
(1): (711-87) / 2 - 43 = 269.
(2): (87+43) / 2 = 65.
(3): (87+711+43+269) / 2 = 555.

Ie. if we call the sequence a, then we have
a(4) = (a(2)-a(1))/2 - a(3),
a(5) = (a(1)+a(3))/2,
a(6) = (a(1)+a(2)+a(3)+a(4))/2.

We can then guess that this pattern repeats modulo 6, so that
a(10) = (a(8)-a(7))/2 - a(9) = (421-39)/2 - 17 = 174,
a(11) = (a(7)+a(9))/2 = (39+17)/2 = 28,
a(12) = (a(7)+a(8)+a(9)+a(10))/2 = (39+421+17+174)/2 = 325.5.

This seems unsatisfactory:
1. a(12) is not an integer.
2. It does not explain a(6k+1), a(6k+2), a(6k+3).

Perhaps pattern (1) is not the right way to define a(6k+4).