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Decimal expansion of inf{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = f(n-1,x) + 1 if n is in A000201, else f(n,x) = 1/f(n-1,x).
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%I #19 Nov 10 2024 06:06:17

%S 3,6,6,3,0,4,6,9,4,6,5,3,2,7,2,6,5,6,6,8,2,4,9,4,1,3,1,4,2,9,0,9,6,6,

%T 9,2,9,9,8,4,2,7,8,8,9,3,9,2,5,4,3,1,6,0,4,1,0,3,1,0,3,8,0,6,3,6,0,0,

%U 5,6,4,5,2,9,0,6,1,5,4,6,1,6,9,4,9,5

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

%C 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 = inf{f(n,1)}. The continued fraction of c is [0,2,1,2,1,2,2,1,2,2,1,2, ...], and the continued fraction of sup{f(n,x)}, alias -2 + 1/c, appears to be identical to the Hofstadter eta-sequence at A006340: (2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2,...). Other limiting constants are similarly obtained using other pairs of Beatty sequences:

%C ...

%C Beatty sequence .... inf{f(n,1)} ... sup{f(n,1)}

%C A000201 (tau) ...... A245215 ....... A245216

%C A001951 (sqrt(2)) .. A245217 ....... A245218; cont. fr. A245219

%C A022838 (sqrt(3)) .. A245220 ....... A245221; cont. fr. A245222

%C A054385 (e/(e-1)) .. A245223 ....... A245224; cont. fr. A245225

%H Clark Kimberling, <a href="/A245215/b245215.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n)*(2 + sup{f(n,1)}) = 1.

%F Equals 1/A245216 = A246129 - 2. - _Hugo Pfoertner_, Nov 10 2024

%e c = 0.366304694653272656682494131429096692998... 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}; min(S(12)) = 7/19 = 0.36842...

%t 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 m = Min[N[Table[s[n], {n, 1, 4000}], 300]]

%t t = RealDigits[m] (* A245215 *)

%t (* _Peter J. C. Moses_, Jul 04 2014 *)

%Y Cf. A226080 (infinite Fibonacci tree), A006340, A245216, A245217, A245220, A245223, A246129.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Jul 13 2014