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A128775
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a(n) = denominator of r(n): r(1)=1, r(n+1) = [b(1,n);b(2,n),...,b(n,n)], a continued fraction of rational terms, where {b(k,n)} is the permutation of the first n terms of {r(k)} such that r(n+1) is minimized.
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4
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1, 1, 1, 3, 18, 303, 352995, 118129644339, 10630462562660994073419, 537130892590606689570311912256259162899177699, 1097615007655912703144392745665304220777182546269636437581535437226324546980771706185955699
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OFFSET
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1,4
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LINKS
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EXAMPLE
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The first 5 terms of {r(k)} are: 1,1,2,4/3,25/18. The continued fraction, whose terms are the permutation of the first 5 terms of {r(k)} which leads to the smallest r(6), is [1;2,1,25/18,4/3] = 416/303.
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MAPLE
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Ltoc := proc(L) numtheory[nthconver](L, nops(L)-1) ; end: r := proc(n) option remember ; local m, rL, rp, L ; if n = 1 then 1; else rL := [seq(procname(i), i=1..n-1)] ; rp := combinat[permute](rL) ; m := Ltoc(rL) ; for L in rp do m := min(m, Ltoc(L)) ; od: m ; fi; end: A128775 := proc(n) denom(r(n)) ; end: for n from 1 do printf("%d, \n", A128775(n)) ; od: # R. J. Mathar, Jul 30 2009
tor:= proc(l) local j; infinity; for j from nops(l) to 1 by -1 do l[j]+1/% od end: hs:= proc(l) local ll, h, s, m; ll:= []; h:= nops(l); s:= 1; m:= s; while s<=h do ll:= [ll[], l[m]]; if m=h then h:= h-1; m:= s else s:= s+1; m:= h fi od; ll end: r:= proc(n) option remember; local j; `if`(n=1, 1, tor(hs(sort([seq(r(j), j=1..n-1)])))) end: a:= n-> denom(r(n)): seq(a(n), n=1..12); # Alois P. Heinz, Aug 04 2009
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MATHEMATICA
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r[1] = r[2] = 1;
r[n_] := r[n] = FromContinuedFraction /@ Permutations[Array[r, n-1]] // Min;
Table[a[n] = Denominator[r[n]]; Print[n, " ", a[n]]; a[n], {n, 1, 11}] (* Jean-François Alcover, May 20 2020 *)
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CROSSREFS
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KEYWORD
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frac,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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