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

[Henry in Rotherhithe]

There are also the repeating 

S_1 =	A011655 (0,1,1,0,1,1,...)
S_2 = A007877 (0,1,2,1,0,1,2,1,...)
S_3 =	A078070(unsigned) (0,1,3,4,3,1,0,1,3,4,3,1,...)

> -----Original Message-----
> From: r.rosenthal at web.de 
> Sent: 13 March 2004 22:46
> To: seqfan at ext.jussieu.fr
> Subject: What next in 0,1,6,25,96,361,1350,5041?
> 
> 
> Another way of putting the question above:
> 
> What have the following sequences in common:
> 
> S_4   = A000290    the squares
> S_5   = A004146    Alternate Lucas Numbers - 2
> S_7   = A054493    A Pellian-related sequence
> S_8   = A001108    a(n)-th triangular number is a square
> S_9   = A049684    F(2n)^2 where F() = Fibonacci numbers
> S_20  = A049683    a(n)=(L(6n)-2)/16, L=Lucas Sequence
> S_25  = A089927*   Expansion of 1/((1-x^2)(1-5x+x^2))
> S_36  = A001110    Both triangular and square
> S_49  = A049682    a(n)=(L(8n)-2)/45, L=Lucas sequence
> S_144 = A004191^2  a(n)=S(n,12) (Chebyshev's poly 2. kind)
> 
> where A089927*(n) = A089927(2n-2)   
> 
> 
> Answer: for any three successive members a, b, c we have
>         a * c = (b-1)^2, i.e. the obey the recurrence
> 
>      ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>         a(n-1) * a(n+1) = ( a(n) - 1 )^2
>      ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
> 
>