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%I #117 Nov 15 2023 14:38:59
%S 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,
%T 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,
%U 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
%N Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the k-th Fibonacci number A000045(k).
%C André-Jeannin proved that this constant is irrational.
%C This constant does not belong to the quadratic number field Q(sqrt(5)) (Bundschuh and Väänänen, 1994). - _Amiram Eldar_, Oct 30 2020
%D Daniel Duverney, Number Theory, World Scientific, 2010, 5.22, pp.75-76.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 358.
%H Kenny Lau, <a href="/A079586/b079586.txt">Table of n, a(n) for n = 1..10000</a> (First 1000 terms computed by Joerg Arndt)
%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 308:19 (1989), pp. 539-541.
%H Richard André-Jeannin, <a href="https://fq.math.ca/Scanned/29-1/advanced29-1.pdf">Problem H-450</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 29, No. 1 (1991), p. 89; <a href="https://www.fq.math.ca/Scanned/30-2/advanced30-2.pdf">Comparable</a>, Solution to Problem H-450 by Paul S. Bruckman, ibid., Vol. 30, No. 2 (1992), p. 191-192.
%H Richard André-Jeannin, <a href="https://www.fq.math.ca/Scanned/29-3/andre-jeannin2.pdf">Sequences of Integers Satisfying Recurrence Relations</a>, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 205-208;
%H Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).
%H Paul S. Bruckman, <a href="https://fq.math.ca/Scanned/25-3/elementary25-3.pdf">Problem B-602</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 25, No. 3 (1987), p. 279; <a href="https://www.fq.math.ca/Scanned/26-3/elementary26-3.pdf">Fibonacci Infinite Series</a>, Solution to Problem B-602 by C. Georghiou, ibid., Vol. 26, No. 3 (1988), pp. 281-282.
%H Peter Bundschuh and Keijo Väänänen, <a href="https://eudml.org/doc/90288">Arithmetical investigations of a certain infinite product</a>, Compositio Mathematica, Vol. 91, No. 2 (1994), pp. 175-199.
%H Daniel Duverney, <a href="https://doi.org/10.1007/s000170050008">Irrationalité de la somme des inverses de la suite Fibonacci</a>, Elemente der Mathematik, Vol. 52, No. 1 (1997), pp. 31-36.
%H William Gosper, <a href="http://dspace.mit.edu/handle/1721.1/6088">Acceleration of Series</a>, Artificial Intelligence Memo #304 (1974).
%H W. E. Greig, <a href="https://www.fq.math.ca/Scanned/15-1/greig2.pdf">Sums of Fibonacci reciprocals</a>, The Fibonacci Quarterly, Vol. 15, No. 1 (1977), pp. 46-48.
%H Peter Griffin, <a href="https://www.fq.math.ca/Scanned/30-2/griffin.pdf"> Acceleration of the Sum of Fibonacci Reciprocals</a>, The Fibonacci Quarterly, Vol. 30, No. 2 (1992), pp. 179-181.
%H Sarah H. Holliday and Takao Komatsu, <a href="https://web.archive.org/web/20150911130411/http://www.westga.edu/~integers/a9int2009/a9int2009.pdf">On the sum of reciprocal generalized Fibonacci numbers</a>, Integers 11A (2011), Article 11. Alternate <a href="https://doi.org/10.1515/integ.2011.031">link</a>.
%H A. F. Horadam, <a href="http://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 Fredrik Johansson, <a href="https://fredrikj.net/blog/2023/11/the-reciprocal-fibonacci-constant/">The reciprocal Fibonacci constant</a>.
%H Paul Kinlaw, Michael Morris, and Samanthak Thiagarajan, <a href="https://www.researchgate.net/publication/350886459_SUMS_RELATED_TO_THE_FIBONACCI_SEQUENCE">Sums related to the Fibonacci sequence</a>, Husson University (2021).
%H Tapani Matala-Aho and Marc Prévost, <a href="https://web.archive.org/web/20161020172636/http://cc.oulu.fi/~tma/TAPANI20.pdf">Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers</a>, Ramanujan J., Vol. 11 (2006), pp. 249-261.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html">Reciprocal Fibonacci Constant</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant">Reciprocal Fibonacci constant</a>.
%F Alternating series representation: 3 + Sum_{k >= 1} (-1)^(k+1)/(F(k)*F(k+1)*F(k+2)). - _Peter Bala_, Nov 30 2013
%F From _Amiram Eldar_, Oct 04 2020: (Start)
%F Equals sqrt(5) * Sum_{k>=0} (1/(phi^(2*k+1) - 1) - 2*phi^(2*k+1)/(phi^(4*(2*k+1)) - 1)), where phi is the golden ratio (A001622) (Greig, 1977).
%F Equals sqrt(5) * Sum_{k>=0} (-1)^k/(phi^(2*k+1) - (-1)^k) (Griffin, 1992).
%F Equals A153386 + A153387. (End)
%F From _Gleb Koloskov_, Sep 14 2021: (Start)
%F Equals 1 + c1*(c2 + 32*Integral_{x=0..infinity} f(x) dx),
%F where c1 = sqrt(5)/(8*log(phi)) = A002163/(8*A002390),
%F c2 = 2*arctan(2)+log(5) = 2*A105199+A016628,
%F phi = (1+sqrt(5))/2 = A001622,
%F f(x) = sin(x)*(4+cos(2*x))/((exp(Pi*x/log(phi))-1)*(2*cos(2*x)+3)*(7-2*cos(2*x))) (End)
%F From _Amiram Eldar_, Jan 27 2022: (Start)
%F Equals 3 + 2 * Sum_{k>=1} 1/(F(2*k-1)*F(2*k+1)*F(2*k+2)) (Bruckman, 1987).
%F Equals 2 + Sum_{k>=1} 1/A350901(k) (André-Jeannin, Problem H-450, 1991).
%F Equals lim_{n->oo} A350903(n)/(A350904(n)*A350902(n)) (André-Jeannin, 1991). (End)
%F Equals sqrt(5/4)*Sum_{j>=1} i^(1-j)/sin(j*c) where c = Pi/2 + i*arccsch(2). - _Peter Luschny_, Nov 15 2023
%e 3.35988566624317755317201130291892717968890513373...
%p Digits := 120: c := Pi/2 + I*arccsch(2):
%p Jeannin := n -> sqrt(5/4)*add(I^(1-j)/sin(j*c), j = 1..n):
%p evalf(Jeannin(1000)); # _Peter Luschny_, Nov 15 2023
%t 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 *)
%t 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 *)
%o (PARI) /* Fast computation without splitting into even and odd indices, see the Arndt reference */
%o lambert2(x, a, S)=
%o {
%o /* Return G(x,a) = Sum_{n>=1} a*x^n/(1-a*x^n) (generalized Lambert series)
%o computed as Sum_{n=1..S} x^(n^2)*a^n*( 1/(1-x^n) + a*x^n/(1-a*x^n) )
%o As series in x correct up to order S^2.
%o We also have G(x,a) = Sum_{n>=1} a^n*x^n/(1-x^n) */
%o return( sum(n=1,S, x^(n^2)*a^n*( 1/(1-x^n) + a*x^n/(1-a*x^n) ) ) );
%o }
%o inv_fib_sum(p=1, q=1, S)=
%o {
%o /* Return Sum_{n>=1} 1/f(n) where f(0)=0, f(1)=1, f(n) = p*f(n-1) + q*f(n-1)
%o computed using generalized Lambert series.
%o Must have p^2+4*q > 0 */
%o my(al,be);
%o \\ Note: the q here is -q in the Horadam paper.
%o \\ The following numerical examples are for p=q=1:
%o al=1/2*(p+sqrt(p^2+4*q)); \\ == +1.6180339887498...
%o be=1/2*(p-sqrt(p^2+4*q)); \\ == -0.6180339887498...
%o return( (al-be)*( 1/(al-1) + lambert2(be/al, 1/al, S) ) ); \\ == 3.3598856...
%o }
%o default(realprecision,100);
%o S = 1000; /* (be/al)^S == -0.381966^S == -1.05856*10^418 << 10^-100 */
%o inv_fib_sum(1,1,S) /* 3.3598856... */ /* _Joerg Arndt_, Jan 30 2011 */
%o (PARI) suminf(k=1, 1/(fibonacci(k))) \\ _Michel Marcus_, Feb 19 2019
%o (Sage) m=120; numerical_approx(sum(1/fibonacci(k) for k in (1..10*m)), digits=m) # _G. C. Greubel_, Feb 20 2019
%Y Cf. A000045, A000796, A001622, A002163, A002390, A084119, A093540, A016628, A105199, A153386, A153387.
%Y Cf. A350901, A350902, A350903, A350904.
%K cons,nonn
%O 1,1
%A _Benoit Cloitre_, Jan 26 2003
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