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A268107 Decimal expansion of 'lambda', a Somos quadratic recurrence constant mentioned by Steven Finch. 2
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 (list; constant; graph; refs; listen; history; text; internal format)
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.

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

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Last modified December 8 17:01 EST 2021. Contains 349596 sequences. (Running on oeis4.)