What next in 0,1,6,25,96,361,1350,5041?

[benoit]

> So the general recurrence seems to be
> S_k: a(n) - (k-1)a(n-1) + (k-1)a(n-2) - a(n-3) = 0
>      a(n) = (k-1)a(n-1) - (k-1)a(n-2) + a(n-3)
>
>

There is also the simple 2- order recurrence here :

a(n)=(k-2)*a(n-1)-a(n-2)+2

One can  consider  the 2 complementary families from 
(a(n-1)^r-1)^s/a(n-2) recursions, which generate integer values only 
when r*s is even :

the simple case is r*s=2 producing linear recurrences for 2 types of 
recursion r=1 and s=2 or r=2 and s=1 :

(i) r=1 and s=2 : a(1)=1 a(2)=k  a(n)=(a(n-1)-1)^2/a(n-2) which gives 
the linear relation a(n)=(k-2)*a(n-1)-a(n-2)+2

(ii) r=2 and s=1 : a(1)=1 a(2)=k  a(n)=(a(n-1)^2-1)/a(n-2) which gives  
a(n)=k*a(n-1)-a(n-2)

For ex. (ii) is also related to many  sequence in EIS :

k=3 : A001906
k=4 : A001353
k=5: A004254
k=6 : =A001109
k=7 : A004187
....

Benoit Cloitre


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