Number factorization and a highly suspicous claim

[Gottfried Helms]

Well, the recurrence is not really surprising, as
it is nothing else than the fact, that sqrt of n
can be expressed by the matrix-product of matrices:

 Let C   = {{ 1, a(0) },{ 0,   1 }}
 Let M_k = {{ 0,   1  },{ 1, a(k)}}   // k starting at 1

 Let recursively be for increasing k
   C = C * M_k

 then the nominator of the k'th convergent is C[1,2] and
 the denominator is C[2,2], so, when A() is periodic both
 can be expressed with the same recurrence, with the only
 important difference in their starting terms.

 Since the cf of sqrt(2) is exceptional simple, we have
 that simple recurrence expression for numerator and
 denumerator (which does not mean, that the actual
 factorial decomposition of that nominators and denominators
 were uninteresting...)

Gottfried Helms