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 A093540 Decimal expansion of Sum_{n >= 1} 1/L(n), where L(n) is the n-th Lucas number. 14
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 LINKS G. C. Greubel, Table of n, a(n) for n = 1..10000 Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 308, No. 19 (1989), pp. 539-541. Paul S. Bruckman, Problem B-603, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 25, No. 3 (1987), p. 280; Lucas Analogue, Solution to Problem B-603 by C. Georghiou, ibid., Vol. 26, No. 3 (1988), p. 282. A. F. Horadam, Elliptic functions and Lambert series in the summation of reciprocals in certain recurrence-generated sequences, The Fibonacci Quarterly, Vol. 26, No. 2 (May-1988), pp. 98-114. Yohei Tachiya, Irrationality of certain Lambert series, Tokyo Journal of Mathematics, Vol. 27, No. 1 (2004), pp. 75-85. FORMULA From Amiram Eldar, Oct 04 2020: (Start) Equals Sum_{k>=0} 1/(phi^(2*k+1) - (-1)^k), where phi is the golden ratio (A001622). Equals A153415 + A153416. (End) 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 EXAMPLE 1.96285817320964578286879512867518352664959301716221... MATHEMATICA RealDigits[Sum[1/LucasL[n], {n, 2000}], 10, 120][[1]] (* Harvey P. Dale, Jan 15 2012 *) PROG (PARI) suminf(n=1, 1/(fibonacci(n-1)+fibonacci(n+1))) \\ Charles R Greathouse IV, Jan 15 2012 CROSSREFS Cf. A001622, A000204, A079586, A153415, A153416. Sequence in context: A019961 A327996 A357762 * A252837 A198573 A068925 Adjacent sequences: A093537 A093538 A093539 * A093541 A093542 A093543 KEYWORD nonn,cons AUTHOR Eric W. Weisstein, Jan 04 2004 STATUS approved

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Last modified September 10 21:37 EDT 2024. Contains 375795 sequences. (Running on oeis4.)