A245219 - OEIS

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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

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OFFSET

0,1

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.

LINKS

Table of n, a(n) for n=0..85.

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)

EXAMPLE

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].

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 *)

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

Sequence in context: A101406 A363445 A363348 * A097509 A095206 A344129

Adjacent sequences: A245216 A245217 A245218 * A245220 A245221 A245222

KEYWORD

nonn,cofr,easy

AUTHOR

Clark Kimberling, Jul 13 2014

EXTENSIONS

Offset changed by Andrew Howroyd, Jul 07 2024

STATUS

approved