# [seqfan] Re: Large recurrences with small coefficients

Richard Guy rkg at cpsc.ucalgary.ca
Mon Mar 21 19:49:34 CET 2011

```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)
>
>
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>
>
>

```