A268107 Decimal expansion of 'lambda', a Somos quadratic recurrence constant mentioned by Steven Finch.

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, 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, 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

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

Table of n, a(n) for n=0..126.

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

Cf. A045761, A085835, A112302.

Sequence in context: A083352 A243350 A091559 * A306963 A084762 A188444

Adjacent sequences: A268104 A268105 A268106 * A268108 A268109 A268110

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