A245215 -id:A245215 - cp's OEIS frontend

A245219 Continued fraction expansion of the constant c in A245218; c = sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A001951, else f(n,x) = 1/x.

3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2

Offset: 0

Clark Kimberling, Jul 13 2014

See Comments at A245215.
Likely a duplicate of A097509. - R. J. Mathar, Jul 21 2014
Theorem: Referring to Problem B6 in the 81st William Lowell Putnam Mathematical Competition (see link), in the notation of the first solution, the sequence {c_i} equals A245219. This proves the conjecture in the previous comment. - Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Sep 09 2021.

			c = 3.43648484... ; the first 12 numbers f(n,1) comprise S(12) = {1, 2, 3, 1/3, 4/3, 7/3, 3/7, 10/7, 17/7, 24/7, 7/24, 31/24}; max(S(12)) = 24/7, with continued fraction [3,2,3].

Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Solutions to the 81st William Lowell Putnam Mathematical Competition
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A226080 (infinite Fibonacci tree), A245217, A245218 (decimal expansion), A245222, A245225.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = Sqrt[2]; 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]] (* A245217 *)
ContinuedFraction[max, 120] (* A245219 *)

Offset changed by Andrew Howroyd, Jul 07 2024

A245217 Decimal expansion of inf{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A001951, else f(n,x) = 1/x.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 13 2014

See Comments at A245215.

			c = 0.29099502708659063074051166818377765138543201...  The first 12 numbers f(n,1) comprise S(12) = {1, 2, 3, 1/3, 4/3, 7/3, 3/7, 10/7, 17/7, 24/7, 7/24, 31/24}; min(S(12)) = 7/24 = 0.29166...

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

Cf. A226080 (infinite Fibonacci tree), A245215, A245218, A245220, A245223.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = Sqrt[2]; 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]  (* A245217 *)
(* Peter J. C. Moses, Jul 04 2014 *)

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

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.

Original entry on oeis.org

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

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.

A245223 Decimal expansion of inf{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.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			c = 0.36930639645301230597278169368719066944...  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}; min(S(16)) = 17/46 = 0.36956... and max(S(12)) = 46/17 = 2.7058...

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

Cf. A226080 (infinite Fibonacci tree), A245215, A245217, A245220, A245224.

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]]]; $RecursionLimit = tmpRec;
m = Min[N[Table[s[n], {n, 1, 4000}], 300]]
RealDigits[m]  (* A245223 *)
(* Peter J. C. Moses, Jul 04 2014 *)

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

A245218 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A001951, else f(n,x) = 1/x.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 13 2014

See Comments at A245215.

			c = 3.43648484309813517846105390392471356500...  The first 12 numbers f(n,1) comprise S(12) = {1, 2, 3, 1/3, 4/3, 7/3, 3/7, 10/7, 17/7, 24/7, 7/24, 31/24}; max(S(12)) = 24/7 = 3.42857...

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

Cf. A226080 (infinite Fibonacci tree), A245215, A245217, A245219, A245223.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = Sqrt[2]; 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 = Max[N[Table[s[n], {n, 1, 4000}], 300]]
t = RealDigits[m]  (* A245217 *)
(* Peter J. C. Moses, Jul 04 2014 *)

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

A245221 Decimal expansion of 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.

Original entry on oeis.org

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

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, A245220, 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 = Max[N[Table[s[n], {n, 1, 4000}], 300]]
t = RealDigits[m]  (* A245221 *)
(* Peter J. C. Moses, Jul 04 2014 *)

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

A245224 Decimal expansion of 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.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			c = 2.7077787160050781243402066659631316299233...  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}; min(S(16)) = 17/46 = 0.36956... and max(S(12)) = 46/17 = 2.7058...

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

Cf. A226080 (infinite Fibonacci tree), A245215, A245217, A245220, A245224.

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]]]; $RecursionLimit = tmpRec;
m = Max[N[Table[s[n], {n, 1, 4000}], 300]]
RealDigits[m]  (* A245224 *)
(* Peter J. C. Moses, Jul 04 2014 *)

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

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.

Original entry on oeis.org

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, 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, 0, 6, 8, 8, 6, 4, 9, 3, 2, 6, 4, 2, 8, 6, 8, 0, 0, 6

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 = 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.

			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...

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

Cf. A226080 (infinite Fibonacci tree), A006340, A245215, A245217, A245220, A245223.

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 = Max[N[Table[s[n], {n, 1, 4000}], 300]]
RealDigits[m]  (* A245216 *)
(* Peter J. C. Moses, Jul 04 2014 *)

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

A245225 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.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			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].

Cf. A226080 (infinite Fibonacci tree), A245217, A245219, A245222, A245224 (decimal expansion).

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 *)
ContinuedFraction[max, 120] (* A245225 *)

Offset changed by Andrew Howroyd, Jul 07 2024

cp's OEIS Frontend

A245219 Continued fraction expansion of the constant c in A245218; c = sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A001951, else f(n,x) = 1/x.

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A245217 Decimal expansion of inf{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A001951, else f(n,x) = 1/x.

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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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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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A245223 Decimal expansion of inf{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.

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A245218 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A001951, else f(n,x) = 1/x.

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A245221 Decimal expansion of 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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