cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 13 2014

Keywords

Comments

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.

Examples

			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].

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

Offset changed by Andrew Howroyd, Jul 07 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

Views

Author

Clark Kimberling, Jul 13 2014

Keywords

Comments

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

Examples

			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...

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

a(n)*(2 + sup{f(n,1)}) = 1.
Equals 1/A245216 = A246129 - 2. - Hugo Pfoertner, Nov 10 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

Views

Author

Clark Kimberling, Jul 13 2014

Keywords

Comments

See Comments at A245215.

Examples

			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...

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

a(n)*sup{f(n,1)} = 1.
Showing 1-3 of 3 results.

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