A085835 Decimal expansion of Grossman's constant.

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

Offset: 0

Author

Eric W. Weisstein, Jul 05 2003

Keywords

• nonn
• cons

Comments

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.

From Jon E. Schoenfield, May 09 2014: (Start)

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)

Examples

			0.737338303369284964205595712487438717934551857465797864769389146671411949653...
		

References

• 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.

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 0..100
• Jean-François Alcover, Grossman's sequence graphics
• Eric Weisstein's World of Mathematics, Grossman's Constant

Programs

Mathematica

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 *)

Extensions

Extended to 25 digits by Jean-François Alcover, Apr 02 2014

More digits from Jon E. Schoenfield, May 09 2014