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

benoit abcloitre at wanadoo.fr
Sun Mar 14 00:41:26 CET 2004

```You have also in the same vein :

A081068 which satisfy : a(n+1)a(n-1)=(a(n)+1)^2

Coming back to what you said, it is like a reminiscence of Somos
sequences  :

In fact if I'm not wrong, for any r,s integers >0 such that r*s is even
,  the recursion :

a(1)=1 a(2)>2 and a(n+1)=(a(n)^r-1)^s/a(n-1) is always integer valued.

In particular :

a(1)=1 a(2)=m>2 gives rise to many integer sequences satisfying linear
recurrence. And it is not a surprise to find many of them in the
database.

You can also add terms to the recursion :

a(1)=1 a(2)=1 a(3)>2 and a(n+1)=(a(n-1)*a(n)^r-1)^s/a(n-2) is always
integer valued.

a(1)=1 a(2)=1  a(3)=1 a(4)>2 and
a(n+1)=(a(n-2)*a(n-1)*a(n)^r-1)^s/a(n-3) is always integer valued.

Laurent phenomenon or something like that. Maybe someone can explain
this fact better than me.

Benoit Cloitre

> 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
>      ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>
> All S_r sequences start with 0, 1, r.
>
> I will enter sequence S_6 in the next few days, which is
> the first S_r not in the OEIS.
>
> But A002531 is closely related to S_6, as can be seen from
> the following table:
>
>      S_6     A002531^2   2*A002531^2-2
> ----------------------------------------
>        0                        0
>        1           1
>        6                        6
>       25          25
>       96                       96
>      361         361
>     1350                     1350
>     5041        5041
>    18816                    18816
>    70225       70225
>   262086                   262086
>
> I think it is a fine observation that so many sequences in
> the OEIS obey the same recurrence rule  a c = (b-1)^2.
> And funny enough this rule is never mentioned for any of
> these sequences! Shall I add this as a remark for them?
>
> I recall that this recurrence has been stated some days ago
> by "zaphod" alias Colin Dickson for the squaretriangles, i.e.
> for sequence A001110.
>
> Let me finish my lengthy posting with the answer to the
> question in the subject line: 18816 (see table).
>
> Best regards
> Rainer Rosenthal
> r.rosenthal at web.de
>
>
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