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

Author

Jean-François Alcover, Jan 26 2016

Keywords

• cons
• nonn

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.

Examples

			0.39952466709679946552503347433225833221736985467599689773670052894853...
		

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

Crossrefs

Cf. A045761, A085835, A112302.

Programs

Magma

// See the link to Jon E. Schoenfield's program.

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]]

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.

Extensions

Extended to 127 digits using Jon E. Schoenfield's evaluation, Aug 27 2016