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%I #44 Jan 05 2025 19:51:37
%S 1,9,6,2,8,5,8,1,7,3,2,0,9,6,4,5,7,8,2,8,6,8,7,9,5,1,2,8,6,7,5,1,8,3,
%T 5,2,6,6,4,9,5,9,3,0,1,7,1,6,2,2,1,9,4,2,1,1,3,0,7,1,5,2,4,0,4,1,7,0,
%U 6,1,6,0,7,5,4,6,4,6,0,3,7,7,9,7,9,0,4,1,8,9,9,0,8,4,0,3,4,6,9,6,2,2
%N Decimal expansion of Sum_{n >= 1} 1/L(n), where L(n) is the n-th Lucas number.
%C André-Jeannin (1989) proved that this constant is irrational, and Tachiya (2004) proved that it does not belong to the quadratic number field Q(sqrt(5)). - _Amiram Eldar_, Oct 30 2020
%H G. C. Greubel, <a href="/A093540/b093540.txt">Table of n, a(n) for n = 1..10000</a>
%H Richard André-Jeannin, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5686125p/f9.image">Irrationalité de la somme des inverses de certaines suites récurrentes</a>, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 308, No. 19 (1989), pp. 539-541.
%H Paul S. Bruckman, <a href="https://fq.math.ca/Scanned/25-3/elementary25-3.pdf">Problem B-603</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 25, No. 3 (1987), p. 280; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/26-3/elementary26-3.pdf">Lucas Analogue</a>, Solution to Problem B-603 by C. Georghiou, ibid., Vol. 26, No. 3 (1988), p. 282.
%H A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/26-2/horadam.pdf">Elliptic functions and Lambert series in the summation of reciprocals in certain recurrence-generated sequences</a>, The Fibonacci Quarterly, Vol. 26, No. 2 (May-1988), pp. 98-114.
%H Yohei Tachiya, <a href="https://projecteuclid.org/euclid.tjm/1244208475">Irrationality of certain Lambert series</a>, Tokyo Journal of Mathematics, Vol. 27, No. 1 (2004), pp. 75-85.
%F From _Amiram Eldar_, Oct 04 2020: (Start)
%F Equals Sum_{k>=0} 1/(phi^(2*k+1) - (-1)^k), where phi is the golden ratio (A001622).
%F Equals A153415 + A153416. (End)
%F Equals 7/3 - 10 * Sum_{k>=1} 1/(L(2*k-1)*L(2*k+1)*L(2*k+2)) (Bruckman, 1987). - _Amiram Eldar_, Jan 27 2022
%e 1.96285817320964578286879512867518352664959301716221...
%t RealDigits[Sum[1/LucasL[n],{n,2000}],10,120][[1]] (* _Harvey P. Dale_, Jan 15 2012 *)
%o (PARI) suminf(n=1,1/(fibonacci(n-1)+fibonacci(n+1))) \\ _Charles R Greathouse IV_, Jan 15 2012
%Y Cf. A001622, A000204, A079586, A153415, A153416.
%K nonn,cons
%O 1,2
%A _Eric W. Weisstein_, Jan 04 2004