[seqfan] Re: Recurrences for continued fractions of sqrt(k)?

[Jan Ritsema van Eck]

Dear seqfans,

Following up on Georg’s question below, I have a question about A003285 (period of the continued fraction of the square root of n, or 0 if n is a square). Here, the same generating function and linear recurrence are mentioned: 

Apparently the generating function of the sequence for the denominators of continued fraction convergents to sqrt(n) is always rational and of form p(x)/[1 - C*x^m + x^(2m)], or equivalently, the denominators satisfy the linear recurrence b(n+2m) = C*b(n+m) - b(n).

This is clear and is identical to the statement in A006702. Then it continues:
 
If so, then it seems that a(n) is half the value of m for each nonsquare n, or 0.

I am not sure if I understand what is meant here. It seems to me that m in the generating function and linear recurrence is meant to to be the period of the continued fraction of sqrt(n). But in that case, the statement that "a(n) is half the value of m" does not make sense, since a(n) is by the definition of the sequence equal to the period of the continued fraction of sqrt(n) and so to m. Did I misinterpret the meaning of m, or am I missing something else? Or can we delete the second sentence as incorrect?

Thanks for any help,
Jan


 

> Op 19 feb. 2021, om 18:48 heeft Georg.Fischer <georg.fischer at t-online.de> het volgende geschreven:
> 
> Dear Seqfans,
> 
> the OEIS has a large collection of the - periodic - continued
> fractions of sqrt(n): periods, numerators/denominators of convergents,
> period lengths etc. (cf. <https://oeis.org/wiki/Index_to_continued_fractions_for_sqrt(n)>)
> 
> For most numerator/denominator sequences the signature of
> a linear recurrence with constant coefficients is given, but
> in <https://oeis.org/A041555> for example it is only conjectured.
> 
> In general, A006702 states:
> "*Apparently* the generating function of the sequence of the denominators of continued fraction convergents to sqrt(n) is always rational and of form p(x)/[1 - C*x^m + x^(2m)], or equivalently, the denominators satisfy the linear recurrence b(n+2m) = C*b(n+m) - b(n). If so, then *it seems* that a(n) is half the value of C for each nonsquare n, or 1. See A003285 for the conjecture regarding m. The same conjectures apply to the sequences of the numerators of continued fraction convergents to sqrt(n)."
> 
> My question is this conjecture - I'm rather convinced that it
> should be true.
> 
> A shorter example shows the pattern:
> 
> A041552	Numerators of continued fraction convergents to sqrt(294).
> 17, 103, 120, 583, 703, *4801*, 163937, 988423 ...
> Period of CF: 17;6,1,4,1,6,34 (cf. my table in A308778)
> Recurrence signature: (0,0,0,0,0,*9602*,0,0,0,0,0,-1)
> 
> I checked a set of cases; the central element C of the signature
> is always 2*numerator(m), where m is the period length, as
> highlighted in the example.
> 
> I suppose that there is somebody in our group how says
> immediately: Oh, trivial, this is known because ...?
> 
> Then we could remove these uncertainities in A006702,
> A003285 etc., and maybe the central terms C could even
> end up in a new (derived) sequence.
> 
> Best regards
> Georg
> 
> 
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