Trouble when computing A006743

[Dean Hickerson]

Neil asked:

> About a week ago there were several postings about trouble
> with the recurrences for A006743.  The current entry gives
> three recurrences stated by Simon Plouffe.  I have labeled
> all three "May be incorrect, needs checking".  Can anyone
> tell me me more about them?  Are /any/ of them correct,
> or should they all be deleted?

Let

    g(x) = (1-2x+x^2-x^4-x^2 sqrt(1-4x^2))/((1+x)^2 (1-2x)^2)

be the generating function given by Paul Zimmermann.

A Mathematica calculation yields this differential equation for g:

                   2       3      4       6
    (2 - 2 x - 12 x  + 12 x  + 4 x  + 16 x ) g(x) +

                2      3       4       5       6       7
       (-x + 2 x  + 7 x  - 12 x  - 16 x  + 16 x  + 16 x ) g'(x) =

                            2       3      4      6
            = 2 - 2 x - 12 x  + 16 x  - 2 x  - 4 x

Because the offset of the sequence is 3, we have

                          n
    g(x)  =  SUM  a(n+3) x
             n>=0
and
                                 n
    g'(x)  =  SUM  (n+1) a(n+4) x .
              n>=0

Rewriting the d.e. in terms of a(n) gives this recurrence:

    (n-5) a(n) = (2n-10) a(n-1) + (7n-47) a(n-2) + (84-12n) a(n-3) +

                 (116-16n) a(n-4) + (16n-128) a(n-5) + (16n-128) a(n-6)

for n>=10.

I wondered if there was something simpler, so I solved the equations
(c0 n+d0)a(n) + ... + (c6 n+d6)a(n-6) = 0  (100 <= n <= 113) for c0, ..., d6.
Only constant multiples of the above recurrence were found.  So unless
something very surprising happens for larger n, this is the simplest
recurrence of this type.

In a separate message to Neil, I'm submitting this comment:

%F A006743 For n>=10, (n-5) a(n) = (2n-10) a(n-1) + (7n-47) a(n-2) + (84-12n) a(n-3) + (116-16n) a(n-4) + (16n-128) a(n-5) + (16n-128) a(n-6). This follows from the differential equation (2-2x-12x^2+12x^3+4x^4+16x^6) g(x) + (-x+2x^2+7x^3-12x^4-16x^5+16x^6+16x^7) g'(x) = 2-2x-12x^2+16x^3-2x^4-4x^6 satisfied by the g.f. sum_n>=0 a(n+3) x^n = (1-2x+x^2-x^4-x^2 sqrt(1-4x^2))/((1+x)^2 (1-2x)^2). (from Dean Hickerson (dean(AT)math.ucdavis.edu), Oct 26 2005)

Dean Hickerson
dean at math.ucdavis.edu