2-dimensional version of Fibonacci numbers

[benoit]

> The orders of the first seven row recurrences are 2 2 4 5 9 11 21.  
> From
> this I don't expect a very easy formula for the diagonal values, but
> perhaps the linked paper contains something magic...

a(n)= A006506(n)

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? Closed form?

B Cloitre










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