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A085835
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Decimal expansion of Grossman's constant.
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2
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7, 3, 7, 3, 3, 8, 3, 0, 3, 3, 6, 9, 2, 8, 4, 9, 6, 4, 2, 0, 5, 5, 9, 5, 7, 1, 2, 4, 8, 7, 4, 3, 8, 7, 1, 7, 9, 3, 4, 5, 5, 1, 8, 5, 7, 4, 6, 5, 7, 9, 7, 8, 6, 4, 7, 6, 9, 3, 8, 9, 1, 4, 6, 6, 7, 1, 4, 1, 1, 9, 4, 9, 6, 5, 3, 2, 3, 3, 9, 3, 7, 2, 7, 5, 4, 3, 9
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OFFSET
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0,1
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COMMENTS
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This is the unique x such that the sequence s_0=1, s_1=x and s_n = s_{n-2}/(1+s_{n-1}) for n >= 2 converges.
It appears that, for large values of n,
s_n -> Sum_{j>=1, k=1..j} c_{j,k} (log(n))^k / n^j,
where c_{j,1}=2 for all j, and where, for each {j,k} such that j > 2 and k > 1, the identity s_n = s_{n-2}/(1+s_{n-1}) can be used to solve for c_{j,k} as a function of c_{2,2}, whose value turns out to be -5.48314176694425549877688093621019843045825... . (End)
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REFERENCES
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S. R. Finch, "Grossman's Constant", Section 6.4 in Mathematical Constants, Cambridge University Press, pp. 429-430, 2003.
Grossman, J. W. "Problem 86-2." Math. Intel. 8, 31, 1986.
Janssen, A. J. E. M. and Tjaden, D. L. A. Solution to Problem 86-2. Math. Intel. 9, 40-43, 1987.
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LINKS
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EXAMPLE
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0.737338303369284964205595712487438717934551857465797864769389146671411949653...
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MATHEMATICA
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digits = 25; precis = 100; m0 = 2*10^6; dm = 10^6; ddm = 3; Clear[var]; var[m_, x0_?NumericQ ] := var[m, x0] = Module[{a, b, n}, Clear[s]; s[0, _] = 1; s[1, x_] := s[1, x] = x; s[n_, x_] := s[n, x] = SetPrecision[s[n-2, x]/(1+s[n-1, x]), precis]; Do[s[n, x0], {n, 1, m}]; fit[n_] = (model = a*n + b) /. FindFit[Table[{k, s[k, x0]}, {k, m, m + ddm}], model, {a, b}, n, WorkingPrecision -> precis]; residuals = Table[s[n, x0] - fit[n], {n, m, m + ddm}]; Variance[residuals]]; Clear[g]; g[m_ /; m < m0] = 3/4; g[m_] := g[m] = Module[{x}, Print["m = ", m]; x /. Last @ FindMinimum[var[m, x], {x, g[m - dm]}, WorkingPrecision -> precis, AccuracyGoal -> digits+1, PrecisionGoal -> digits+1, StepMonitor :> Print["Step to x = ", x]]]; g[m0]; g[m = m0 + dm]; While[RealDigits[g[m], 10, digits] != RealDigits[g[m - dm], 10, digits], m = m + dm]; RealDigits[g[m], 10, digits] // First (* Jean-François Alcover, Apr 02 2014 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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