A338304 Decimal expansion of Sum_{k>=0} 1/L(2^k), where L(k) is the k-th Lucas number (A000032).

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

Offset: 1

Author

Amiram Eldar, Oct 21 2020

Keywords

• nonn
• cons

Comments

Erdős and Graham (1980) asked whether this constant is irrational or transcendental.

Badea (1987) proved that it is irrational, and André-Jeannin (1991) proved that it is not a quadratic irrational in Q(sqrt(5)), in contrast to the corresponding sum with Fibonacci numbers, Sum_{k>=0} 1/F(2^k) = (7-sqrt(5))/2 (A079585).

Bundschuh and Pethö (1987) proved that it is transcendental.

Examples

			1.49792038099906271987068555399285960807207719857085...
		

Links

• Richard André-Jeannin, A note on the irrationality of certain Lucas infinite series, The Fibonacci Quarterly, Vol. 29, No. 2 (1991), pp. 132-136.
• Catalin Badea, The irrationality of certain infinite series, Glasgow Mathematical Journal, Vol. 29, No. 2 (1987), pp. 221-228.
• Peter Bundschuh and Attila Pethö, Zur transzendenz gewisser Reihen, Monatshefte für Mathematik, Vol. 104, No. 3 (1987), pp. 199-223, alternative link.
• Paul Erdős and Ronald L. Graham, Old and new problems and results in combinatorial number theory, L'enseignement Mathématique, Université de Genève, 1980, pp. 64-65.
• Index entries for transcendental numbers

Crossrefs

Cf. A000045, A000032, A001566, A079585, A338305.

Programs

Mathematica

RealDigits[Sum[1/LucasL[2^n], {n, 0, 10}], 10, 100][[1]]

Formula

Equals 1 + Sum_{k>=0} 1/A001566(k).