A245221 -id:A245221 - cp's OEIS frontend

A245222 Continued fraction expansion of the constant c in A245221; c = sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1

Offset: 0

Clark Kimberling, Jul 14 2014

See Comments at A245215.

Appears to be the same as the sequence 1 + [x == 0 (mod sqrt(3))], as x runs over the elements of N U N*sqrt(3) in increasing order, where N = {0, 1, 2, 3, ...} and [...] is the Iverson bracket. - M. F. Hasler, Feb 06 2025

			c = 2.7207664507294752975469517348171513242... ; the first 12 numbers f(n,1) comprise S(12) = {1, 2, 1/2, 3/2, 2/3, 5/3, 8/3, 3/8, 11/8, 8/11, 19/11, 11/19}; max(S(12)) = 8/3, with continued fraction [2,1,2].
From _M. F. Hasler_, Feb 06 2025: (Start)
Illustration of the "multiple of sqrt(3)" comment:
  n:  0   1   2    3   4    5    6   7   8    9   10  11  12   13  14  15
  x:  0   1  1.73  2   3   3.46  4   5  5.20  6  6.93  7   8  8.66  9  10
  m:  1   0   1    0   0    1    0   0   1    0   1    0   0   1    0   0
Here, x lists the elements of N U N*sqrt(3), and m = 1 if x == 0 (mod sqrt(3)), i.e., x is an integer multiple of sqrt(3). The sequence a(n) is m + 1. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1999

Cf. A226080 (infinite Fibonacci tree), A245217, A245219, A245220, A245221 (decimal expansion).
Cf. A144612 (Sturmian word of slope (3-sqrt(3))/2; same as 2-a(n)).
Cf. A006337 (Hofstadter's eta sequence: an analog with sqrt(2)).

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = Sqrt[3]; 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]] (* A245221 *)
ContinuedFraction[max, 120] (* A245222 *)

/* illustration of the comment related to sqrt(3)*/
[1+(abs(x-x\/s*s)<1e-9) | x<-Set(concat(Col([1.,s=sqrt(3)]~*[0..99])))[1..99] ] \\ M. F. Hasler, Feb 06 2025

a(n) = 2 - A144612(n) for all n > 0. - M. F. Hasler, Feb 06 2025

Offset changed by Andrew Howroyd, Aug 08 2024

A245215 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).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 13 2014

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:
...
Beatty sequence .... inf{f(n,1)} ... sup{f(n,1)}
A000201 (tau) ...... A245215 ....... A245216
A001951 (sqrt(2)) .. A245217 ....... A245218; cont. fr. A245219
A022838 (sqrt(3)) .. A245220 ....... A245221; cont. fr. A245222
A054385 (e/(e-1)) .. A245223 ....... A245224; cont. fr. A245225

			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...

Clark Kimberling, Table of n, a(n) for n = 1..1000

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

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;
m = Min[N[Table[s[n], {n, 1, 4000}], 300]]
t = RealDigits[m]  (* A245215 *)
(* Peter J. C. Moses, Jul 04 2014 *)

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

Equals 1/A245216 = A246129 - 2. - Hugo Pfoertner, Nov 10 2024

A245220 Decimal expansion of inf{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			c = 0.367543491184951248721260972541092540...  The first 12 numbers f(n,1) comprise S(12) = {1, 2, 1/2, 3/2, 2/3, 5/3, 8/3, 3/8, 11/8, 8/11, 19/11, 11/19}; min(S(12)) = 3/8 = 0.375... and max(S(12)) = 8/3 = 2.666...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A226080 (infinite Fibonacci tree), A245215, A245217, A245221, A245222.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = Sqrt[3]; 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;
m = Min[N[Table[s[n], {n, 1, 4000}], 300]]
t = RealDigits[m]  (* A245220 *)
(* Peter J. C. Moses, Jul 04 2014 *)

a(n)*sup{f(n,1)} = 1.

cp's OEIS Frontend

A245222 Continued fraction expansion of the constant c in A245221; c = sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

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A245215 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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A245220 Decimal expansion of inf{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

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