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%I #53 Mar 31 2024 12:04:53
%S 3,9,9,5,2,4,6,6,7,0,9,6,7,9,9,4,6,5,5,2,5,0,3,3,4,7,4,3,3,2,2,5,8,3,
%T 3,2,2,1,7,3,6,9,8,5,4,6,7,5,9,9,6,8,9,7,7,3,6,7,0,0,5,2,8,9,4,8,5,3,
%U 0,7,3,7,0,2,7,1,2,5,9,3,4,5,6,6,3,4,9,2,0,9,8,2,0,2,5,7,5,7,4,3,3,3,1,0,0,1,6,6,7,5,1,6,4,6,9,6,4,7,4,5,6,4,4,9,8,4,5
%N Decimal expansion of 'lambda', a Somos quadratic recurrence constant mentioned by Steven Finch.
%C [Quoted from _Steven Finch_] Another Somos constant lambda = 0.3995246670... arises as follows: If k < lambda, then the sequence h_0 = 0, h_1 = k, h_n = h_(n-1)*(1 + h_(n-1) - h_(n-2)) for n>=2 converges to a limit less than 1; if k > lambda, then the sequence diverges to infinity. This is similar to Grossman's constant.
%C A heuristical evaluation of lambda = 0.39952466709679946552503347433225833221736985467599... was communicated to me by _Jon E. Schoenfield_ in a private email.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic Recurrence Constants, p. 446.
%H Jon E. Schoenfield, <a href="/A268107/a268107.txt">Magma program</a> communicated to J.-F. Alcover.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/SomossQuadraticRecurrenceConstant.html">Somos's Quadratic Recurrence Constant</a>
%F Conjecture: lambda is the radius of convergence of the function Sum_{n>=0} A045761(n)*x^n, that is the constant 1/d computed by _Vaclav Kotesovec_ in A045761.
%e 0.39952466709679946552503347433225833221736985467599689773670052894853...
%t n0 (* initial number of terms *) = 2*10^7; iter = 10^5; dn = 10^6; k1 = 0.3; k2 = 0.4; eps = 10^-16; f[k_?NumericQ] := (h0 = 0; h1 = k; h2 = k*(1+k); Do[h0 = h1; h1 = h2; h2 = Min[h1 + (h1-h0), h1*(1+h1-h0)], {iter}]; h2); Clear[g]; g[n0] = k1; g[n = n0+dn] = k2; g[n_] := g[n] = k /. FindRoot[f[k]==1, {k, g[n-dn] }]; While[Print[n, " ", g[n] // RealDigits]; Abs[g[n] - g[n-dn]] > eps, n = n+dn]; lambda = g[n]; RealDigits[lambda][[1]][[1;;9]]
%o (Magma) // See the link to _Jon E. Schoenfield_'s program.
%Y Cf. A045761, A085835, A112302.
%K cons,nonn
%O 0,1
%A _Jean-François Alcover_, Jan 26 2016
%E Extended to 127 digits using _Jon E. Schoenfield_'s evaluation, Aug 27 2016
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