A055060 Decimal expansion of Komornik-Loreti constant.

1, 7, 8, 7, 2, 3, 1, 6, 5, 0, 1, 8, 2, 9, 6, 5, 9, 3, 3, 0, 1, 3, 2, 7, 4, 8, 9, 0, 3, 3, 7, 0, 0, 8, 3, 8, 5, 3, 3, 7, 9, 3, 1, 4, 0, 2, 9, 6, 1, 8, 1, 0, 9, 9, 7, 7, 8, 4, 7, 8, 1, 4, 7, 0, 5, 0, 5, 5, 5, 7, 4, 9, 1, 7, 5, 0, 6, 0, 5, 6, 8, 6, 9, 9, 1, 3, 1, 0, 0, 1, 8, 6, 3, 4, 0, 7, 5, 3, 3, 3, 0, 2

Offset

1,2

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 438-439.

Links

Table of n, a(n) for n=1..102.

Pieter Allaart, and Derong Kong, On the smallest base in which a number has a unique expansion, arXiv:2006.07927 [math.NT], 2020.

Jean-Paul Allouche and Michel Cosnard, The Komornik-Loreti constant is transcendental, Amer. Math. Monthly, Vol. 107, No. 5 (May, 2000), pp. 448-449, preprint.

Vilmos Komornik and Paola Loreti, Unique Developments in Non-Integer Bases, Amer. Math. Monthly, Vol. 105, No. 7 (Aug. - Sep., 1998), pp. 636-639.

Vilmos Komornik and Derong Kong, Bases with two expansions, arXiv:1705.00473 [math.NT], 2017.

Vilmos Komornik, Wolfgang Steiner, and Yuru Zou, Unique double base expansions, arXiv:2209.02373 [math.NT], 2022.

Eric Weisstein's World of Mathematics, Komornik-Loreti Constant.

Index entries for transcendental numbers

Formula

This number q (say) is defined by 1 = Sum_{n >= 1} A010060(n)/q^n.

The unique positive solution of the equation Product_{k>=0} (1 - 1/q^(2^k)) = (2-q)/(q-1) (Allouche and Cosnard, 2000). - Amiram Eldar, Oct 22 2020

Example

1.787231650...

Mathematica

First[ RealDigits[q /. FindRoot[ Sum[ Mod[ DigitCount[n, 2, 1], 2]/q^n, {n, 1, 2000}] == 1, {q, 1.8}, WorkingPrecision -> 120], 10, 102]](* Jean-François Alcover, Jun 11 2012, after PARI *)

Prog

(PARI) solve(q=1.7, 1.8, sum(n=1, 2000, (subst(Pol(binary(n)), x, 1)%2)/q^n)-1)

Crossrefs

The continued-fraction expansion of this number is in A080890.

Cf. A010060.

Sequence in context: A020506 A193345 A197822 * A010515 A021131 A220610

Adjacent sequences: A055057 A055058 A055059 * A055061 A055062 A055063

Keyword

nonn,cons,easy

Author

N. J. A. Sloane, Jun 11 2000

Extensions

More terms from Ralf Stephan, Mar 30 2003

Status

approved