A289252 Decimal expansion of the mean number of iterations in a comparison algorithm using centered continued fractions, a constant related to Vallée's constant.

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

Offset

1,3

Comments

From Jon E. Schoenfield, Jan 27 2018: (Start)

If we define the partial sum s_k = (360/Pi^4) * Sum_{i..k} Sum_{j=ceiling(phi*i)..floor((phi+1)*i)} 1/(i^2*j^2), then the real-valued sequence s_0, s_1, s_2, s_3, ... converges very slowly, and the convergence is not smooth because of the aperiodicity created by the presence of the functions ceiling(phi*i) and floor((phi+1)*i) in the limits on j in the inner sum. However, if we define the partial sum S_k = s_Fibonacci(k), then the real-valued sequence S_0, S_1, S_2, S_3, ... converges fairly quickly. (Cf. A228639.)

Also, the subsequences S_Even = {s_0, s_1, s_3, s_8, s_21, ..., s_Fibonacci(2*d), ...} for d >= 0 and S_Odd = {s_1, s_2, s_5, s_13, s_34, ..., s_Fibonacci(2*d+1), ...} for d >= 0 both converge to lim_{k->infinity} s_k = 1.08922147... in a way that can be accelerated using successive applications of Richardson extrapolation, and--given the values of s_Fibonacci(m) for m=0..27--appears to yield the limit 1.08922147386406851032218345320... (This would seem to indicate that the last two terms currently in the Data section are incorrect.) (End)

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.19 Vallée's constant, p. 162.

Links

Table of n, a(n) for n=1..12.

Eric Weisstein's MathWorld, Vallée Constant

Formula

(360/Pi^4) * Sum_{i >= 1} Sum_{j=ceiling(phi*i)..floor((phi+1)*i)} 1/(i^2*j^2).

Example

1.0892214740...

Mathematica

terms = 10^6;
f[i_Integer] := f[i] = NSum[1/(i^2*j^2), {j, Ceiling[ GoldenRatio * i], Floor[(1 + GoldenRatio) * i]}, WorkingPrecision -> 30];
s = 360/Pi^4 * NSum[f[i], {i, 1, Infinity}, Method -> "WynnEpsilon", NSumTerms -> terms];
RealDigits[s, 10, 12][[1]] (* updated Jun 14 2019 *)

Crossrefs

Cf. A074903, A143302.

Sequence in context: A030167 A010767 A334751 * A064734 A090929 A113210

Adjacent sequences: A289249 A289250 A289251 * A289253 A289254 A289255

Keyword

nonn,cons,more

Author

Jean-François Alcover, Sep 02 2017

Extensions

Corrected and extended to 12 digits by Jean-François Alcover, Jun 14 2019, after Jon E. Schoenfield's pertinent comment.

Status

approved