2-dimensional version of Fibonacci numbers

[Brendan McKay]

* benoit <abcloitre at wanadoo.fr> [031116 12:35]:
> 
> limit n-->infty  (a(n))^(1/n^2) =c1=1.50304..  is the Hard Square 
> Entropy Constant A085850 with no known closed form formula (to be 
> compared with the amazing formula for the Hard Hexagon Entropy Constant 
>  http://mathworld.wolfram.com/HardHexagonEntropyConstant.html.) 
>  
> It seems there is another possible constant for this sequence a(n) : 
> 
> letting b(n)=a(n+2)*a(n)/a(n+1)^2 
> 
>  limit n-->infty b(n)=c2= 2.2591... 
> 
> with  b(2n)<c2<b(2n-1) 
> 
> Any more accurate value for this limit c2?

It is almost certainly the square of c1.  Perhaps this can be proved
from the known asymptotic behaviour of a(n), but I'm not sure.

My experimental observation is that a(n) appears to behave like
  A * c3^n * c1^(n^2) 
where c1 is as above, 
  c3 = 1.143519129587 approx
   A = 1.0660826 approx

I base this conjecture on numerical analysis of a(n) for n up to 19.

Brendan.