A074903 -id:A074903 - cp's OEIS frontend

A247318 Decimal expansion of p_2, a probability associated with continuant polynomials.

0, 4, 8, 4, 8, 0, 8, 0, 1, 4, 4, 9, 4, 6, 3, 6, 3, 2, 7, 0, 5, 7, 2, 4, 9, 3, 3, 8, 8, 2, 4, 7, 6, 5, 5, 6, 3, 3, 3, 0, 5, 6, 0, 0, 6, 6, 9, 5, 2, 3, 7, 1, 3, 9, 7, 7, 1, 6, 6, 5, 5, 9, 9, 8, 3, 8, 6, 6, 2, 0, 4, 8, 2, 0, 5, 4, 0, 2, 2, 5, 4, 2, 7, 6, 2, 5, 8, 8, 8, 8, 8, 7, 3, 1, 1, 3, 3, 9, 2, 4, 7, 7

Offset: 0

Jean-François Alcover, Sep 12 2014

			0.04848080144946363270572493388247655633305600669523713977...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.19 Vallée's Constant, p. 161.

Wikipedia, Continuant polynomial.

Cf. A074903, A145426.

digits = 101; s = NSum[(-1)^n*(n + 1)*Zeta[n + 4]*(Zeta[n + 2] - 1), {n, 0, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits + 10]; p2 = -5 + 2*Pi^2/3 - 2*Zeta[3] + 2*s; Join[{0}, RealDigits[p2, 10, digits] // First]

p_2 = Sum_{i >= 1}(sum_{j >= 1} 1/((i*j + 1)^2*(i*j + i + 1)^2)).

p_2 = Sum_{n >= 0} (-1)^n*(n + 1)*zeta(n + 4)*(zeta(n + 2) - 1).

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

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 02 2017

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)

			1.08922147386...

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

Eric Weisstein's World of Mathematics, Vallée Constant.

Cf. A074903, A143302.

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 *)

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

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

cp's OEIS Frontend

A143303 Duplicate of A074903.

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A247318 Decimal expansion of p_2, a probability associated with continuant polynomials.

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

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