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

Author

Jean-François Alcover, Sep 02 2017

Keywords

• nonn
• cons
• more

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)

Examples

			1.08922147386...
		

References

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

Links

• Eric Weisstein's World of Mathematics, Vallée Constant.

Crossrefs

Cf. A074903, A143302.

Programs

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 *)

Formula

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

Extensions

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