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

[Jan Ritsema van Eck]

Hallo Georg,

Did anyone reply to your question yet?
I think the conjecture is correct, with a small correction. If m is odd, the recurrence seems to be b(n+2m)=C*b(n+m)+b(n). This is obvious for m=1, since b(n+2)=a(n+2)*b(n+1)+b(n) with a(n+2) the n+2nd coefficient in the continued fraction.
I think I have a proof for the conjecture, let me know if you are still interested.

Best regards,

Jan Ritsema van Eck

> 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
> 
> 
> --
> Seqfan Mailing list - http://list.seqfan.eu/