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Decimal expansion of Grossman's constant.
2

%I #57 Jul 14 2019 22:01:30

%S 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,

%T 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,

%U 1,9,4,9,6,5,3,2,3,3,9,3,7,2,7,5,4,3,9

%N Decimal expansion of Grossman's constant.

%C 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.

%C From _Jon E. Schoenfield_, May 09 2014: (Start)

%C It appears that, for large values of n,

%C s_n -> Sum_{j>=1, k=1..j} c_{j,k} (log(n))^k / n^j,

%C 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)

%D S. R. Finch, "Grossman's Constant", Section 6.4 in Mathematical Constants, Cambridge University Press, pp. 429-430, 2003.

%D Grossman, J. W. "Problem 86-2." Math. Intel. 8, 31, 1986.

%D Janssen, A. J. E. M. and Tjaden, D. L. A. Solution to Problem 86-2. Math. Intel. 9, 40-43, 1987.

%H Jon E. Schoenfield, <a href="/A085835/b085835.txt">Table of n, a(n) for n = 0..100</a>

%H Jean-François Alcover, <a href="/A085835/a085835.gif">Grossman's sequence graphics</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GrossmansConstant.html">Grossman's Constant</a>

%e 0.737338303369284964205595712487438717934551857465797864769389146671411949653...

%t 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 *)

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Jul 05 2003

%E Extended to 25 digits by _Jean-François Alcover_, Apr 02 2014

%E More digits from _Jon E. Schoenfield_, May 09 2014