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Decimal expansion of Sum_{k>=0} 1/L(2^k), where L(k) is the k-th Lucas number (A000032).
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%I #12 Jul 28 2024 13:02:23

%S 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,

%T 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,

%U 5,2,7,7,6,9,5,3,2,2,3,7,8,3,9,9,3,2,1

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

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

%C 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).

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

%H Richard André-Jeannin, <a href="https://www.fq.math.ca/Scanned/29-2/andre-jeannin.pdf">A note on the irrationality of certain Lucas infinite series</a>, The Fibonacci Quarterly, Vol. 29, No. 2 (1991), pp. 132-136.

%H Catalin Badea, <a href="https://doi.org/10.1017/S0017089500006868">The irrationality of certain infinite series</a>, Glasgow Mathematical Journal, Vol. 29, No. 2 (1987), pp. 221-228.

%H Peter Bundschuh and Attila Pethö, <a href="https://doi.org/10.1007/BF01547953">Zur transzendenz gewisser Reihen</a>, Monatshefte für Mathematik, Vol. 104, No. 3 (1987), pp. 199-223, <a href="https://eudml.org/doc/178351">alternative link</a>.

%H Paul Erdős and Ronald L. Graham, <a href="http://www.math.ucsd.edu/~fan/ron/papers/80_11_number_theory.pdf">Old and new problems and results in combinatorial number theory</a>, L'enseignement Mathématique, Université de Genève, 1980, pp. 64-65.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

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

%e 1.49792038099906271987068555399285960807207719857085...

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

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

%K nonn,cons

%O 1,2

%A _Amiram Eldar_, Oct 21 2020