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 A085835 #58 Feb 16 2025 08:32:50
%S A085835 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 A085835 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 A085835 1,9,4,9,6,5,3,2,3,3,9,3,7,2,7,5,4,3,9
%N A085835 Decimal expansion of Grossman's constant.
%C A085835 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 A085835 From _Jon E. Schoenfield_, May 09 2014: (Start)
%C A085835 It appears that, for large values of n,
%C A085835 s_n -> Sum_{j>=1, k=1..j} c_{j,k} (log(n))^k / n^j,
%C A085835 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 A085835 S. R. Finch, "Grossman's Constant", Section 6.4 in Mathematical Constants, Cambridge University Press, pp. 429-430, 2003.
%D A085835 Grossman, J. W. "Problem 86-2." Math. Intel. 8, 31, 1986.
%D A085835 Janssen, A. J. E. M. and Tjaden, D. L. A. Solution to Problem 86-2. Math. Intel. 9, 40-43, 1987.
%H A085835 Jon E. Schoenfield, <a href="/A085835/b085835.txt">Table of n, a(n) for n = 0..100</a>
%H A085835 Jean-François Alcover, <a href="/A085835/a085835.gif">Grossman's sequence graphics</a>
%H A085835 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GrossmansConstant.html">Grossman's Constant</a>
%e A085835 0.737338303369284964205595712487438717934551857465797864769389146671411949653...
%t A085835 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 A085835 nonn,cons
%O A085835 0,1
%A A085835 _Eric W. Weisstein_, Jul 05 2003
%E A085835 Extended to 25 digits by _Jean-François Alcover_, Apr 02 2014
%E A085835 More digits from _Jon E. Schoenfield_, May 09 2014