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%I #5 Jul 20 2014 00:15:18
%S 2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,1,
%T 2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,1,2,2,
%U 2,1,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2
%N Continued fraction expansion of the constant c in A245224; c = sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A054385, else f(n,x) = 1/x.
%C See Comments at A245215.
%e c = 2.70777871600507812434020666596313162... ; The first 16 numbers f(n,1) comprise S(16) = {1, 2, 1/2, 3/2, 5/2, 2/5, 7/5, 12/5, 5/12, 17/12, 12/17, 29/17}; max(S(16)) = 46/17, with continued fraction [2, 1, 2, 2, 2].
%t tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = E/(E-1); w = Table[Floor[k*r], {k, 2000}]; s[1] = 1; s[n_] := s[n] = If[MemberQ[w, n - 1], u[s[n - 1]], d[s[n - 1]]]; max = Max[N[Table[s[n], {n, 1, 3000}], 200]] (* A245224 *)
%t ContinuedFraction[max, 120] (* A245225 *)
%Y Cf. A226080 (infinite Fibonacci tree), A245217, A245219, A245222, A245224.
%K nonn,cofr,easy
%O 1,1
%A _Clark Kimberling_, Jul 14 2014
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