OFFSET
0,1
COMMENTS
[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.
A heuristical evaluation of lambda = 0.39952466709679946552503347433225833221736985467599... was communicated to me by Jon E. Schoenfield in a private email.
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic Recurrence Constants, p. 446.
LINKS
Steven R. Finch, A Deceptively Simple Quadratic Recurrence, arXiv:2409.03510 [math.NT], 2024. See p. 9.
Jon E. Schoenfield, Magma program communicated to J.-F. Alcover.
Eric Weisstein's MathWorld, Somos's Quadratic Recurrence Constant
FORMULA
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.
EXAMPLE
0.39952466709679946552503347433225833221736985467599689773670052894853...
MATHEMATICA
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]]
PROG
(Magma) // See the link to Jon E. Schoenfield's program.
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Jean-François Alcover, Jan 26 2016
EXTENSIONS
Extended to 127 digits using Jon E. Schoenfield's evaluation, Aug 27 2016
STATUS
approved