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 A079586 Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the k-th Fibonacci number A000045(k). 18
 3, 3, 5, 9, 8, 8, 5, 6, 6, 6, 2, 4, 3, 1, 7, 7, 5, 5, 3, 1, 7, 2, 0, 1, 1, 3, 0, 2, 9, 1, 8, 9, 2, 7, 1, 7, 9, 6, 8, 8, 9, 0, 5, 1, 3, 3, 7, 3, 1, 9, 6, 8, 4, 8, 6, 4, 9, 5, 5, 5, 3, 8, 1, 5, 3, 2, 5, 1, 3, 0, 3, 1, 8, 9, 9, 6, 6, 8, 3, 3, 8, 3, 6, 1, 5, 4, 1, 6, 2, 1, 6, 4, 5, 6, 7, 9, 0, 0, 8, 7, 2, 9, 7, 0, 4 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS André-Jeannin proved that this constant is irrational. LINKS Kenny Lau, Table of n, a(n) for n = 1..10000 (First 1000 terms computed by Joerg Arndt) Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 308:19 (1989), pp. 539-541. Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012). William Gosper, Acceleration of Series, Artificial Intelligence Memo #304 (1974). S. H. Holliday, T. Komatsu, On the sum of reciprocal generalized Fibonacci numbers, Integers 11A (2011) # 11 [Broken link]. Alternate link. 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, pp.98-114, (May-1988). Tapani Matala-Aho and Marc Prévost, Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers, Ramanujan J 11 (2006), pp. 249-261. Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant FORMULA Alternating series representation: 3 + Sum_{k >= 1} (-1)^(k+1)/(F(k)*F(k+1)*F(k+2)). - Peter Bala, Nov 30 2013 EXAMPLE 3.35988566624317755317201130291892717968890513373... MATHEMATICA digits = 105; Sqrt[5]*NSum[(-1)^n/(GoldenRatio^(2*n + 1) - (-1)^n), {n, 0, Infinity}, WorkingPrecision -> digits, NSumTerms -> digits] // RealDigits[#, 10, digits] & // First (* Jean-François Alcover, Apr 09 2013 *) First@RealDigits[Sqrt[5]/4 ((Log[5] + 2 QPolyGamma[1, 1/GoldenRatio^4] - 4 QPolyGamma[1, 1/GoldenRatio^2])/(2 Log[GoldenRatio]) + EllipticTheta[2, 0, 1/GoldenRatio^2]^2), 10, 105] (* Vladimir Reshetnikov, Nov 18 2015 *) PROG (PARI) /* Fast computation without splitting into even and odd indices, see the Arndt reference */ lambert2(x, a, S)= { /* Return G(x, a) = Sum_{n>=1} a*x^n/(1-a*x^n) (generalized Lambert series)    computed as Sum_{n=1..S} x^(n^2)*a^n*( 1/(1-x^n) + a*x^n/(1-a*x^n) ) )    As series in x correct up to order S^2.    We also have G(x, a) = Sum_{n>=1} a^n*x^n/(1-x^n) */     return( sum(n=1, S, x^(n^2)*a^n*( 1/(1-x^n) + a*x^n/(1-a*x^n) ) ) ); } inv_fib_sum(p=1, q=1, S)= { /* Return Sum_{n>=1} 1/f(n) where f(0)=0, f(1)=1, f(n) = p*f(n-1) + q*f(n-1)    computed using generalized Lambert series.    Must have p^2+4*q > 0 */     my(al, be);     \\ Note: the q here is -q in the Horadam paper.     \\ The following numerical examples are for p=q=1:     al=1/2*(p+sqrt(p^2+4*q));  \\ == +1.6180339887498...     be=1/2*(p-sqrt(p^2+4*q));  \\ == -0.6180339887498...     return( (al-be)*( 1/(al-1) + lambert2(be/al, 1/al, S) ) ); \\ == 3.3598856... } default(realprecision, 100); S = 1000; /* (be/al)^S == -0.381966^S == -1.05856*10^418 << 10^-100 */ inv_fib_sum(1, 1, S) /* 3.3598856... */ /* Joerg Arndt, Jan 30 2011 */ CROSSREFS Cf. A000045, A084119, A093540. Sequence in context: A073060 A183526 A087343 * A125960 A141584 A287195 Adjacent sequences:  A079583 A079584 A079585 * A079587 A079588 A079589 KEYWORD cons,nonn,changed AUTHOR Benoit Cloitre, Jan 26 2003 STATUS approved

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Last modified August 17 12:52 EDT 2018. Contains 313817 sequences. (Running on oeis4.)