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A245216 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A000201, else f(n,x) = 1/x.

%I A245216 #4 Jul 17 2014 22:12:05
%S A245216 2,7,2,9,9,6,7,7,4,1,5,9,9,8,0,2,4,8,7,8,9,1,6,4,6,7,7,4,8,7,5,9,0,7,
%T A245216 5,2,1,1,4,3,7,8,4,1,1,3,5,3,7,0,3,4,6,2,5,9,8,6,9,5,2,7,2,4,5,2,9,0,
%U A245216 0,6,8,8,6,4,9,3,2,6,4,2,8,6,8,0,0,6
%N A245216 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A000201, else f(n,x) = 1/x.
%C A245216 Equivalently, f(n,x) = 1/(f(n-1,x) if n is in A001950 (upper Wythoff sequence, given by w(n) = floor[tau*n], where tau = (1 + sqrt(5))/2, the golden ratio) and f(n,x) = f(n-1) + 1 otherwise.  Let c = sup{f(n,1)}.   The continued fraction of c is [2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, ...], which appears to be identical to the Hofstadter eta-sequence at A006340. See Comments at A245215.
%H A245216 Clark Kimberling, <a href="/A245216/b245216.txt">Table of n, a(n) for n = 1..1000</a>
%F A245216 inf{f(n,1)}*(2 + a(n)) = 1.
%e A245216 c = 2.7299677415998024878916467748759075211...  The first 12 numbers f(n,1) comprise S(12) = {1, 2, 1/2, 3/2, 5/2, 2/5, 7/5, 5/7, 12/7, 19/7, 7/19, 26/19}; max(S(12)) = 19/7 = 2.71429...
%t A245216 tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = GoldenRatio; 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]]]; $RecursionLimit = tmpRec;
%t A245216 m = Max[N[Table[s[n], {n, 1, 4000}], 300]]
%t A245216 RealDigits[m]  (* A245216 *)
%t A245216 (* _Peter J. C. Moses_, Jul 04 2014 *)
%Y A245216 Cf. A226080 (infinite Fibonacci tree), A006340, A245215, A245217, A245220, A245223.
%K A245216 nonn,cons,easy
%O A245216 1,1
%A A245216 _Clark Kimberling_, Jul 13 2014