A105216 - OEIS

%I #30 Jul 10 2024 06:30:01

%S 1,2,7,31,164,1021,7340,59899,547423,5541311,61560751,744810564,

%T 9749580487,137299957892,2069988277027,33266800950301,567742165061876,

%U 10254686071781119,195439907769223706,3919618523321600065,82517650453354285621,1819502802723019762607

%N Maximum denominator among the n! ratios equal to the continued fractions which have the permutations of (1,2,3,...,n) for terms.

%C Calculated by Vladeta Jovovic and David W. Wilson.

%H Alois P. Heinz, <a href="/A105216/b105216.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) ~ c * n!, where c = 1.69579254611555585961617066333... . - _Vaclav Kotesovec_, Aug 25 2014

%F From _Mark van Hoeij_, Jul 08 2024: (Start)

%F Conjecture: a(n) = n*a(n-1) + (n-1)*a(n-3) + a(n-4).

%F Consequently: a(n) = round( sqrt(2/Pi) * (exp(-1)*BesselI(1,1)+ sinh(1)*BesselI(0,1)) * BesselK(n/2+1/2,1) * BesselK(n/2+1,1) ). (End)

%F Consequently: c = exp(-1)*BesselI(1,1)+ sinh(1)*BesselI(0,1). - _Vaclav Kotesovec_, Jul 09 2024

%p r:= proc(l) local j; infinity; for j to nops(l) do l[j] +1/% od end: gl:= proc(n) local i, l; l:=[]; for i from 2 to n do l:= `if` (irem (i, 2)=0, [l[], i], [i, l[]]) od; [l[], 1] end: a:= n-> denom (r (gl (n))): seq (a(n), n=1..25);  # _Alois P. Heinz_, Nov 18 2009

%t r[l_] := Module[{j, f = Infinity}, For[j = 1, j <=  Length[l], j++, f = l[[j]] + 1/f]; f];

%t gl[n_] := Module[{i, l = {}}, For[i = 2, i <= n, i++, l = If[Mod [i, 2] == 0, Append[l, i], Prepend[l, i]]]; Append[l, 1]];

%t a[n_] := Denominator [r[gl[n]]];

%t Table[a[n], {n, 1, 25}] (* _Jean-François Alcover_, Nov 18 2020, after _Alois P. Heinz_ *)

%Y Cf. A105151.

%K nonn

%O 1,2

%A _Leroy Quet_, Apr 12 2005

%E More terms from _Alois P. Heinz_, Nov 18 2009