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A079586
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Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the k-th Fibonacci number A000045(k).
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18
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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
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OFFSET
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1,1
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COMMENTS
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André-Jeannin proved that this constant is irrational.
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LINKS
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Joerg Arndt, Table of n, a(n) for n = 1..1000
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
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
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FORMULA
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Alternating series representation: 3 + Sum_{k >= 1} (-1)^(k+1)/(F(k)*F(k+1)*F(k+2)). - Peter Bala, Nov 30 2013
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EXAMPLE
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3.35988566624317755317201130291892717968890513373...
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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Cf. A000045, A084119, A093540.
Sequence in context: A073060 A183526 A087343 * A125960 A141584 A179437
Adjacent sequences: A079583 A079584 A079585 * A079587 A079588 A079589
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KEYWORD
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cons,nonn
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AUTHOR
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Benoit Cloitre, Jan 26 2003
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STATUS
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approved
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