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.
%I A289252 #41 Feb 16 2025 08:33:48
%S A289252 1,0,8,9,2,2,1,4,7,3,8,6
%N A289252 Decimal expansion of the mean number of iterations in a comparison algorithm using centered continued fractions, a constant related to Vallée's constant.
%C A289252 From _Jon E. Schoenfield_, Jan 27 2018: (Start)
%C A289252 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.)
%C A289252 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)
%D A289252 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.19 Vallée's constant, p. 162.
%H A289252 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ValleeConstant.html">Vallée Constant</a>.
%F A289252 Equals (360/Pi^4) * Sum_{i >= 1} Sum_{j=ceiling(phi*i)..floor((phi+1)*i)} 1/(i^2*j^2).
%e A289252 1.08922147386...
%t A289252 terms = 10^6;
%t A289252 f[i_Integer] := f[i] = NSum[1/(i^2*j^2), {j, Ceiling[ GoldenRatio * i], Floor[(1 + GoldenRatio) * i]}, WorkingPrecision -> 30];
%t A289252 s = 360/Pi^4 * NSum[f[i], {i, 1, Infinity}, Method -> "WynnEpsilon", NSumTerms -> terms];
%t A289252 RealDigits[s, 10, 12][[1]] (* updated Jun 14 2019 *)
%Y A289252 Cf. A074903, A143302.
%K A289252 nonn,cons,more
%O A289252 1,3
%A A289252 _Jean-François Alcover_, Sep 02 2017
%E A289252 Corrected and extended to 12 digits by _Jean-François Alcover_, Jun 14 2019, after _Jon E. Schoenfield_'s pertinent comment.