[seqfan] Re: Large recurrences with small coefficients

[Alexander P-sky]

For A188123

g.f. (3 + 2*x + 2*x^3 + 2*x^5 - x^6)/((-1 + x)^4*(1 + 2*x + 2*x^2 +
x^3)) - already found by R. Mathar

a(n) = 1/108*(8*sqrt(3)*sin((2*pi*n)/3)+3*(2*n*(4*n*(n+3)+21)+9*i*sin(pi*n)+35)-24*cos((2*pi*
n)/3)+27*cos(pi*n))

On 3/21/11, Richard Guy <rkg at cpsc.ucalgary.ca> wrote:
> Dear all,
>           I don't understand the definition of these sequences,
> but the characteristic polynomials do indeed factor. Their
> roots are roots of unity, with some repetitions, and are
> most perspicuously expressed, perhaps, as
>
>                  (x - 1)^2 * (x^2 - 1) * (x^3 - 1)
>
>             (x - 1)^2 * (x^2 - 1) * (x^3 - 1) * (x^4 - 1)
>
>        (x - 1)^2 * (x^2 - 1) * (x^3 - 1) * (x^4 - 1) * (x^5 - 1)
>
> and (presumably, but I haven't checked)
>
> (x - 1)^2 * (x^2 - 1) * (x^3 - 1) * (x^4 - 1) * (x^5 - 1) * (x^6 - 1)
>
> The recurrences can be solved to give explicit formulas for the
> members of the sequences.  For example, the first sequence, A188123,
> with a different offset, starting  0, 1, 3, 8, 16, 31, ..., is
> something like
>
>    {8n^3 + 18n + 9[1 - (-1)^n]}/36  -  8{w^n - w^(-n)}/{w^2 - w}
>
> where  w^2 + w + 1 = 0.   Would someone kindlily correct this and
> supply the other formulas?   Thanks!    R.
>
> On Mon, 21 Mar 2011, Ron Hardin wrote:
>
>> Some partition-related empirical linear recurrences that might factor
>>
>> https://oeis.org/A188123
>> a(n)=2*a(n-1)-a(n-3)-a(n-4)+2*a(n-6)-a(n-7)
>>
>> https://oeis.org/A188124
>> a(n)=2*a(n-1)-a(n-3)-2*a(n-5)+2*a(n-6)+a(n-8)-2*a(n-10)+a(n-11)
>>
>> https://oeis.org/A188125
>> a(n)=2*a(n-1)-a(n-3)-a(n-5)+2*a(n-8)-a(n-11)-a(n-13)+2*a(n-15)-a(n-16)
>>
>> https://oeis.org/A188126
>> a(n)=2*a(n-1)-a(n-3)-a(n-5)+a(n-6)-2*a(n-7)+2*a(n-8)+a(n-9)-a(n-13)-2*a(n-14)+2*a(n-15)-a(n-16)+a(n-17)+a(n-19)-2*a(n-21)+a(n-22)
>>
>>
>> https://oeis.org/A188127
>> a(n)=2*a(n-1)-a(n-3)-a(n-5)+a(n-6)-a(n-7)+a(n-9)+a(n-10)+a(n-12)-2*a(n-13)-2*a(n-16)+a(n-17)+a(n-19)+a(n-20)-a(n-22)+a(n-23)-a(n-24)-a(n-26)+2*a(n-28)-a(n-29)
>>
>>
>> tabl https://oeis.org/A188122
>> T(n,k)=Number of strictly increasing arrangements of n nonzero numbers in
>> -(n+k-2)..(n+k-2) with sum zero
>>
>>
>>
>> rhhardin at mindspring.com
>> rhhardin at att.net (either)