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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). 16
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

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

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 *)

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.

Sequence in context: A073060 A183526 A087343 * A125960 A141584 A179437

Adjacent sequences:  A079583 A079584 A079585 * A079587 A079588 A079589

KEYWORD

cons,nonn

AUTHOR

Benoit Cloitre, Jan 26 2003

STATUS

approved

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Last modified October 2 05:40 EDT 2014. Contains 247537 sequences.