Continued Fractions Which Are Permutations

[David Wilson]

In the following sequences, a(n) gives the conjectural maximum and
minimum numerators and denominators for a rational whose continued
fraction may be given by a permutation of (1,...,n) (1 <= n <= 30).

In each case, the first eight elements are accurate, while the remaining
elements are computed from conjectural optimal permutations gotten by
extrapolating patterns found in the optimal permutations for the first eight
elements.

Based on the accurate elements, none of these sequences is in the OEIS.
Please make the obvious conjecture about the relationship between the
minimum numerator and minimum denominator sequences.

maximum numerator
1 3 11 48 253 1576 11331 92467 845064 8554195 95032146 1149773923
15050556403 211951761735 3195468293093 51354400809456 876431092504915
15830294577832786 301703171661686235 6050766978392127541
127383588868883838996 2808790552014917701633 64735601273039395629696
1556590303863908215537153 38982295953434932297107013
1015160893611686239533226371 27449880211017263120396707691
769650724489529974451345241852 22348334497513449908728239459109
671247096612416609578578221522680

minimum numerator
1 3 9 37 183 1089 7507 59261 525432 5185027 56276118 667218665
8572665529 118743064065 1763010417987 27944432899993 470820846422697
8404897200626691 158440099278231667 3145660094900520781
65599808580014388882 1433810922365584581697 32773404612628181853894
781981995659270875907413 19440987778013415886708939
502816281630159621550235139 13508063261577304006721723085
376431933092185971050365020241 10866984702187333531397887909815
324603756514736725727499676141569

maximum denominator
1 2 7 31 164 1021 7340 59899 547423 5541311 61560751 744810564
9749580487 137299957892 2069988277027 33266800950301 567742165061876
10254686071781119 195439907769223706 3919618523321600065
82517650453354285621 1819502802723019762607 41934991510050298965097
1008341621820157645331676 25252291130419746961880089
657609763681467401502772346 17781722436558453989652442015
498571048425779865035884470499 14476998730021827593029689150248
434826289461078704423992098098221

minimum denominator
1 1 3 9 37 183 1089 7507 59261 525432 5185027 56276118 667218665
8572665529 118743064065 1763010417987 27944432899993 470820846422697
8404897200626691 158440099278231667 3145660094900520781
65599808580014388882 1433810922365584581697 32773404612628181853894
781981995659270875907413 19440987778013415886708939
502816281630159621550235139 13508063261577304006721723085
376431933092185971050365020241 10866984702187333531397887909815

----- Original Message ----- 
From: "Leroy Quet" <qq-quet at mindspring.com>
To: <ham>
Cc: "seqfan" <seqfan at ext.jussieu.fr>
Sent: Sunday, April 10, 2005 12:02 PM
Subject: Continued Fractions Which Are Permutations


>I just submitted the following.
>
>>%S A000001 1,3,11,48
>>%N A000001 Greatest numerator among the n! ratios equal to the continued
>>fractions which have the permutations of (1,2,3,...,n) for terms.
>>%e A000001 a(4) = 48 because the continued fractions [4;2,1,3] (= 48/11)
>>and [3;1,2,4] (= 48/13) have the greatest numerators among continued
>>fraction which each have a permutation of (1,2,3,4) for terms.
>>%O A000001 1
>>%K A000001 ,more,nonn,
>
>
> But with only 4 terms, the sequences of least numerators (1,3,9,37,...),
> greatest denominators (1,2,7,31,...), and least denominators
> (1,1,3,9,...) each bring up several hits.
>
> Also the numerators (and probably the denominators as well) of the
> greatest ratios
> (1,3,11/3,47/10,...) and of least ratios (1,3/2,9/7,38/31,...) among CFs
> which are permutations of (1,2,3,..,n) bring up several hits.
>
> Could someone calculate more terms, or are these sequences already in the
> database under different names?
>
> thanks,
> Leroy Quet