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

Author

Clark Kimberling, Jul 14 2014

Keywords

• nonn
• cofr
• easy

Comments

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

Examples

			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)
		

Links

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

Crossrefs

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)).

Programs

Mathematica

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 *)

PARI

/* 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

Formula

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

Extensions

Offset changed by Andrew Howroyd, Aug 08 2024