A093540 -id:A093540 - cp's OEIS frontend

A079586 Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the k-th Fibonacci number A000045(k).

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

Offset: 1

Benoit Cloitre, Jan 26 2003

André-Jeannin proved that this constant is irrational.

This constant does not belong to the quadratic number field Q(sqrt(5)) (Bundschuh and Väänänen, 1994). - Amiram Eldar, Oct 30 2020

			3.35988566624317755317201130291892717968890513373...

Daniel Duverney, Number Theory, World Scientific, 2010, 5.22, pp.75-76.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 358.

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.
Richard André-Jeannin, Problem H-450, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 29, No. 1 (1991), p. 89; Comparable, Solution to Problem H-450 by Paul S. Bruckman, ibid., Vol. 30, No. 2 (1992), p. 191-192.
Richard André-Jeannin, Sequences of Integers Satisfying Recurrence Relations, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 205-208;
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Paul S. Bruckman, Problem B-602, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 25, No. 3 (1987), p. 279; Fibonacci Infinite Series, Solution to Problem B-602 by C. Georghiou, ibid., Vol. 26, No. 3 (1988), pp. 281-282.
Peter Bundschuh and Keijo Väänänen, Arithmetical investigations of a certain infinite product, Compositio Mathematica, Vol. 91, No. 2 (1994), pp. 175-199.
Daniel Duverney, Irrationalité de la somme des inverses de la suite Fibonacci, Elemente der Mathematik, Vol. 52, No. 1 (1997), pp. 31-36.
William Gosper, Acceleration of Series, Artificial Intelligence Memo #304 (1974).
W. E. Greig, Sums of Fibonacci reciprocals, The Fibonacci Quarterly, Vol. 15, No. 1 (1977), pp. 46-48.
Peter Griffin, Acceleration of the Sum of Fibonacci Reciprocals, The Fibonacci Quarterly, Vol. 30, No. 2 (1992), pp. 179-181.
Sarah H. Holliday and Takao Komatsu, On the sum of reciprocal generalized Fibonacci numbers, Integers 11A (2011), Article 11. 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 (May-1988), pp. 98-114.
Fredrik Johansson, The reciprocal Fibonacci constant.
Paul Kinlaw, Michael Morris, and Samanthak Thiagarajan, Sums related to the Fibonacci sequence, Husson University (2021).
Tapani Matala-Aho and Marc Prévost, Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers, Ramanujan J., Vol. 11 (2006), pp. 249-261.
Z.W. M. Trzaska, Fibonacci Polynomials their Properties and Applications, Z. Anal. Anwend. 15 (1996), no. 3, pp. 729-746.
Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.
Wikipedia, Reciprocal Fibonacci constant.

Cf. A000045, A000796, A001622, A002163, A002390, A084119, A093540, A016628, A105199, A153386, A153387.

Cf. A350901, A350902, A350903, A350904.

Digits := 120: c := Pi/2 + I*arccsch(2):
Jeannin := n -> sqrt(5/4)*add(I^(1-j)/sin(j*c), j = 1..n):
evalf(Jeannin(1000)); # Peter Luschny, Nov 15 2023

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 *)

/* 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 */

suminf(k=1, 1/(fibonacci(k))) \\ Michel Marcus, Feb 19 2019

m=120; numerical_approx(sum(1/fibonacci(k) for k in (1..10*m)), digits=m) # G. C. Greubel, Feb 20 2019

Alternating series representation: 3 + Sum_{k >= 1} (-1)^(k+1)/(F(k)*F(k+1)*F(k+2)). - Peter Bala, Nov 30 2013
From Amiram Eldar, Oct 04 2020: (Start)
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).
Equals sqrt(5) * Sum_{k>=0} (-1)^k/(phi^(2*k+1) - (-1)^k) (Griffin, 1992).
Equals A153386 + A153387. (End)
From Gleb Koloskov, Sep 14 2021: (Start)
Equals 1 + c1*(c2 + 32*Integral_{x=0..infinity} f(x) dx),
where c1 = sqrt(5)/(8*log(phi)) = A002163/(8*A002390),
      c2 = 2*arctan(2)+log(5) = 2*A105199+A016628,
      phi = (1+sqrt(5))/2 = A001622,
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)
From Amiram Eldar, Jan 27 2022: (Start)
Equals 3 + 2 * Sum_{k>=1} 1/(F(2*k-1)*F(2*k+1)*F(2*k+2)) (Bruckman, 1987).
Equals 2 + Sum_{k>=1} 1/A350901(k) (André-Jeannin, Problem H-450, 1991).
Equals lim_{n->oo} A350903(n)/(A350904(n)*A350902(n)) (André-Jeannin, 1991). (End)
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
Equals lim_{n->oo} A203006(n)/A003266(n) (Z.W. M. Trzaska, 1996). - Raul Prisacariu, Sep 04 2024

A153415 Decimal expansion of Sum_{n>=1} 1/A000032(2*n).

Original entry on oeis.org

5, 6, 6, 1, 7, 7, 6, 7, 5, 8, 1, 1, 3, 8, 4, 5, 5, 0, 2, 7, 5, 9, 2, 9, 3, 2, 1, 2, 1, 2, 0, 6, 2, 0, 0, 3, 7, 3, 6, 1, 4, 4, 1, 9, 7, 8, 6, 5, 9, 0, 5, 5, 7, 0, 4, 9, 2, 3, 4, 4, 4, 1, 3, 2, 5, 4, 5, 7, 5, 5, 5, 4, 5, 3, 0, 2, 0, 8, 6, 8, 5, 6, 1, 4, 8, 5, 5, 6, 7, 8, 4, 2, 1, 8, 1, 8, 3, 2, 6, 6, 4, 6, 1, 5, 3

Offset: 0

Eric W. Weisstein, Dec 25 2008

From Peter Bala, Oct 15 2019: (Start)
c = (1/4)*(theta_3( (3-sqrt(5))/2 )^2 - 1 ), where theta_3(q) = 1 + 2*Sum_{n >= 1} q^n^2. See Borwein and Borwein, Proposition 3.5 (i), p. 91. Cf. A056854.
Series acceleration formulas (L(n) = A000032(n)):
c = 1 - 5*Sum_{n >= 1} 1/( L(2*n)*(L(2*n)^2 - 5) ).
c = (1/6) + 15*Sum_{n >= 1} 1/( L(2*n)*(L(2*n)^2 + 5) ).
c = (11/16) - 10*Sum_{n >= 1} (L(2*n)^2 - 10)/( L(2*n)*(L(2*n)^2 - 5)*(L(2*n)^2 - 20) ). (End)
Compare with Sum_{n >= 1} 1/(L(2*n) - sqrt(5)) = phi and Sum_{n >= 1} 1/(L(2*n) + sqrt(5)) = 2 - phi, where phi = (sqrt(5) + 1)/2. - Peter Bala, Nov 23 2019
This constant is transcendental (Duverney et al., 1997). - Amiram Eldar, Oct 30 2020

			0.56617767581138455027...

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 91.

Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.
Index entries for transcendental numbers

Cf. A000032, A000122, A056854, A093540, A153416.

First[ RealDigits[ N[(EllipticTheta[3, 0, GoldenRatio^(-2)]^2 - 1)/4, 120], 10, 105]](* Jean-François Alcover, Jun 07 2012, after Eric W. Weisstein *)

th3(x)=1 + 2*suminf(n=1,x^n^2)
phi=(sqrt(5)+1)/2
(th3(phi^-2)^2-1)/4 \\ Charles R Greathouse IV, Jun 06 2016

A304092 Number of Lucas numbers (A000032: 2, 1, 3, 4, 7, 11, ...) dividing n.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 2, 3, 2, 2, 2, 4, 1, 3, 2, 3, 1, 4, 1, 3, 3, 3, 1, 4, 1, 2, 2, 4, 2, 3, 1, 3, 3, 2, 2, 5, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 2, 4, 2, 2, 2, 3, 1, 4, 2, 4, 2, 3, 1, 4, 1, 2, 3, 3, 1, 4, 1, 3, 2, 3, 1, 5, 1, 2, 2, 4, 3, 3, 1, 3, 2, 2, 1, 5, 1, 2, 3, 4, 1, 4, 2, 3, 2, 3, 1, 4, 1, 3, 3, 3, 1, 3, 1, 3, 3

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000032, A093540, A102460, A304091, A304094, A304096.

Cf. also A005086, A300837.

Module[{nn=11,luc},luc=LucasL[Range[0,nn]];Table[Count[n/luc,?IntegerQ],{n,Max[luc]}]] (* _Harvey P. Dale, Jul 01 2023 *)

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304092(n) = sumdiv(n,d,A102460(d));

a(n) = Sum_{d|n} A102460(d).
a(n) = A304091(n) + A102460(n).
a(n) = A304094(n) + A059841(n) = A304096(n) + A059841(n) + A079978(n) + 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 + 1/2 = 2.462858... . - Amiram Eldar, Dec 31 2023

A304096 Number of Lucas numbers larger than 3 (4, 7, 11, 18, ...) that divide n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1

Offset: 1

Antti Karttunen, May 13 2018

nonn

a(n) is the number of the divisors d of n that are of the form d = A000045(k-1) + A000045(k+1), for k >= 3.

			The divisors of 4 are 1, 2 and 4. Of these only 4 is a Lucas number larger than 3, thus a(4) = 1.
The divisors of 28 are 1, 2, 4, 7, 14 and 28. Of these 4 and 7 are Lucas numbers (A000032) larger than 3, thus a(28) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000032, A000045, A000204, A093540, A102460, A304092, A304094, A304095, A304104.

Cf. also A005086, A300837.

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304096(n) = sumdiv(n,d,(d>3)*A102460(d));

a(n) = Sum_{d|n, d>3} A102460(d).
a(n) = A304094(n) - A079978(n) - 1.
a(n) = A304092(n) - A059841(n) - A079978(n) - 1.
a(n) = A007949(A304104(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 - 4/3 = 0.629524... . - Amiram Eldar, Dec 31 2023

A105393 Decimal expansion of sum of reciprocals of squares of Fibonacci numbers.

Original entry on oeis.org

2, 4, 2, 6, 3, 2, 0, 7, 5, 1, 1, 6, 7, 2, 4, 1, 1, 8, 7, 7, 4, 1, 5, 6, 9, 4, 1, 2, 9, 2, 6, 6, 2, 0, 3, 7, 4, 3, 2, 0, 2, 5, 9, 7, 7, 4, 5, 1, 3, 8, 3, 0, 9, 0, 5, 1, 1, 0, 1, 0, 2, 8, 3, 4, 5, 4, 6, 6, 1, 1, 9, 3, 7, 5, 1, 1, 1, 9, 7, 8, 6, 3, 6, 8, 7, 7, 5, 3, 8, 9, 8, 1, 5, 2, 1, 5, 3, 6, 3, 6, 3, 7, 9, 2, 1

Offset: 1

Jonathan Vos Post, Apr 04 2005

Known to be transcendental. - Benoit Cloitre, Jan 07 2006
Compare with Sum_{n >= 1} 1/(F(n)^2 + 1) = (5*sqrt(5) - 3)/6 and Sum_{n >= 3} 1/(F(n)^2 - 1) = (43 - 15*sqrt(5))/18. - Peter Bala, Nov 19 2019
Duverney et al. (1997) proved that this constant is transcendental. - Amiram Eldar, Oct 30 2020

			2.426320751167241187741569...

Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, C. R. Acad. Sci. Paris Ser. I Math., Vol. 308, No. 19 (1989), pp. 539-541.
Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.
Michel Waldschmidt, Elliptic functions and transcendence, in: Krishnaswami Alladi (ed.), Surveys in number theory, Springer, New York, NY, 2008, pp. 143-188, alternative link. See Corollary 51.
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.
Index entries for transcendental numbers

Cf. A000045, A007598 (squares of Fibonacci numbers).

Cf. A079586, A093540, A105394.

RealDigits[Total[1/Fibonacci[Range[500]]^2],10,120][[1]] (* Harvey P. Dale, May 31 2016 *)

sum(k=1,500,1./fibonacci(k)^2) \\ Benoit Cloitre, Jan 07 2006

Equals Sum_{k>=1} 1/F(k)^2 = 2.4263207511672411877... - Benoit Cloitre, Jan 07 2006

More terms from Benoit Cloitre, Jan 07 2006

A304095 a(n) is the number of the proper divisors of n that are Lucas numbers larger than 3 (4, 7, 11, 18, ...).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1

Offset: 1

Antti Karttunen, May 13 2018

nonn

a(n) is the number of the proper divisors d of n that are of the form d = A000045(k-1) + A000045(k+1), for k >= 3.

			The proper divisors of 28 are 1, 2, 4, 7 and 14. Of these 4 and 7 are Lucas numbers (A000032) larger than 3, thus a(28) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000045, A000204, A093540, A102460, A304091, A304093, A304096, A304102.

Cf. also A300836.

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304095(n) = sumdiv(n,d,(d>3)*(dA102460(d));

a(n) = Sum_{d|n, d>3, dA102460(d).
a(n) = A007949(A304102(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 - 4/3 = 0.629524... . - Amiram Eldar, Jul 05 2025

A105394 Decimal expansion of sum of reciprocals of squares of Lucas numbers.

Original entry on oeis.org

1, 2, 0, 7, 2, 9, 1, 9, 9, 6, 9, 8, 5, 7, 4, 7, 0, 7, 4, 4, 1, 7, 2, 0, 4, 1, 8, 4, 2, 5, 7, 6, 9, 9, 9, 4, 5, 3, 0, 6, 9, 2, 1, 4, 5, 4, 0, 1, 9, 0, 3, 6, 3, 7, 6, 9, 5, 1, 3, 1, 1, 5, 9, 4, 2, 2, 1, 2, 2, 4, 0, 0, 1, 5, 4, 0, 7, 0, 3, 5, 7, 7, 6, 1, 6, 7, 7, 6, 5, 5, 9, 7, 8, 6, 8, 8, 9, 9, 9, 2

Offset: 1

Jonathan Vos Post, Apr 04 2005

This constant is transcendental (Duverney et al., 1997). - Amiram Eldar, Oct 30 2020

			1.207291996985747074417204...

Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Wiley, 1987, p. 97.

Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, C. R. Acad. Sci. Paris Ser. I Math., Vol. 308, No. 19 (1989), pp. 539-541.
Paul S. Bruckman,, Problem H-347, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 20, No. (1982), p. 372; It All Adds Up, Solution to Problem H-347 by the proposer, ibid., Vol. 22, No. 1 (1984), pp. 94-96.
Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.
Index entries for transcendental numbers

Cf. A000032, A001254 (squares of Lucas numbers).

Cf. A079586, A093540, A105393, A153415.

f[n_] := f[n] = RealDigits[ Sum[ 1/LucasL[k]^2, {k, 1, n}], 10, 100] // First; f[n=100]; While[f[n] != f[n-100], n = n+100]; f[n] (* Jean-François Alcover, Feb 13 2013 *)

Equals Sum_{n >= 1} 1/L(n)^2.
Equals (1/8)*( theta_3(beta)^4 - 1 ), where beta = (3 - sqrt(5))/2 and theta_3(q) = 1 + 2*Sum_{n >= 1} q^(n^2) is a theta function. See Borwein and Borwein, Exercise 7(f), p. 97. - Peter Bala, Nov 13 2019
Equals c*(2*c+1), where c = A153415 (follows from the identity Sum_{n=-oo..oo} 1/L(n^2) = (Sum_{n=-oo..oo} 1/L(2*n))^2, see Bruckman, 1982). - Amiram Eldar, Jan 27 2022

A153416 Decimal expansion of Sum_{n>=0} 1/A000032(2*n+1).

Original entry on oeis.org

1, 3, 9, 6, 6, 8, 0, 4, 9, 7, 3, 9, 8, 2, 6, 1, 2, 3, 2, 5, 9, 2, 8, 6, 5, 8, 0, 7, 4, 6, 3, 1, 2, 1, 5, 2, 2, 9, 1, 3, 4, 4, 8, 8, 1, 9, 2, 9, 6, 3, 1, 3, 8, 5, 0, 6, 3, 8, 3, 7, 0, 8, 2, 7, 1, 6, 2, 4, 8, 6, 0, 5, 3, 0, 1, 6, 2, 5, 1, 6, 9, 2, 3, 6, 4, 1, 8, 6, 2, 2, 0, 6, 6, 2, 2, 1, 6, 3, 6, 9, 5, 8, 0, 3, 4

Offset: 1

Eric W. Weisstein, Dec 25 2008

			1.3966804973982612325...

Cf. A000032, A002878, A093540, A153415.

RealDigits[ NSum[ 1/LucasL[2*n + 1], {n, 0, Infinity}, WorkingPrecision -> 110, NSumTerms -> 100], 10, 105] // First (* Jean-François Alcover, Feb 07 2013 *)

From Amiram Eldar, Jul 05 2025: (Start)
Equals Sum_{n>=0} 1/A002878(n).
Equals A093540 - A153415. (End)

A228040 Decimal expansion of sum of reciprocals, row 2 of Wythoff array, W = A035513.

Original entry on oeis.org

6, 2, 9, 5, 2, 4, 8, 3, 9, 8, 7, 6, 3, 1, 2, 4, 4, 9, 5, 3, 5, 4, 6, 1, 7, 9, 5, 3, 4, 1, 8, 5, 0, 1, 9, 3, 3, 1, 6, 2, 5, 9, 6, 8, 3, 8, 2, 8, 8, 8, 6, 0, 8, 7, 7, 9, 7, 3, 8, 1, 9, 0, 7, 0, 8, 3, 7, 2, 8, 2, 7, 4, 2, 1, 3, 1, 2, 7, 0, 4, 4, 6, 4, 5, 7, 0

Offset: 0

Clark Kimberling, Aug 05 2013

Let c be the constant given by A079586, that is, the sum of reciprocals of the Fibonacci numbers F(k) for k>=1. The number c-1, the sum of reciprocals of row 1 of W, is known to be irrational (see A079586). Conjecture: the same is true for all the other rows of W.

Let h be the constant given at A153387 and s(n) the sum of reciprocals of numbers in row n of W. Then h < 1 + s(n)*floor(n*tau) < c. Thus, s(n) -> 0 as n -> oo.

			1/4 + 1/7 + 1/11 + ... = 0.629524839876312449535461795341...

Cf. A035513, A079586, A093540, A228041, A228042, A228043.

f[n_] := f[n] = Fibonacci[n]; g = GoldenRatio; w[n_, k_] := w[n, k] = f[k + 1]*Floor[n*g] + f[k]*(n - 1);
n = 2; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
RealDigits[r, 10]

Equals A093540 - 4/3. - Amiram Eldar, May 22 2021

cp's OEIS Frontend

A079586 Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the k-th Fibonacci number A000045(k).

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A153415 Decimal expansion of Sum_{n>=1} 1/A000032(2*n).

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A304092 Number of Lucas numbers (A000032: 2, 1, 3, 4, 7, 11, ...) dividing n.

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A304096 Number of Lucas numbers larger than 3 (4, 7, 11, 18, ...) that divide n.

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A105393 Decimal expansion of sum of reciprocals of squares of Fibonacci numbers.

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A304095 a(n) is the number of the proper divisors of n that are Lucas numbers larger than 3 (4, 7, 11, 18, ...).

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A105394 Decimal expansion of sum of reciprocals of squares of Lucas numbers.

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