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A105151
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Greatest numerator among the n! ratios equal to the continued fractions which have the permutations of (1,2,3,...,n) for terms.
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3
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1, 3, 11, 48, 253, 1576, 11331, 92467, 845064, 8554195, 95032146, 1149773923, 15050556403, 211951761735, 3195468293093, 51354400809456, 876431092504915, 15830294577832786, 301703171661686235, 6050766978392127541, 127383588868883838996, 2808790552014917701633
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) ~ c * n!, where c = 2.61781739197011756877854274... . - Vaclav Kotesovec, Aug 25 2014
Conjecture: a(n) = n*a(n-1) + (n-1)*a(n-3) + a(n-4).
Consequently: a(n) = round( sqrt(2/Pi) * (cosh(1)*BesselI(0,1)+ sinh(1)*BesselI(1,1)) * BesselK(n/2+1/2,1) * BesselK(n/2+1,1) ). (End)
Consequently: c = cosh(1)*BesselI(0,1)+ sinh(1)*BesselI(1,1). - Vaclav Kotesovec, Jul 09 2024
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EXAMPLE
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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.
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MAPLE
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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 to n do l:= `if` (irem (i, 2)=0, [l[], i], [i, l[]]) od; l end: a:= n-> numer (r (gl (n))): seq (a(n), n=1..30); # Alois P. Heinz, Nov 18 2009
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MATHEMATICA
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Table[Max@ Map[Numerator@ FromContinuedFraction@ # &, Permutations@ Range@ n], {n, 10}] (* Michael De Vlieger, Sep 25 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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