A084119 -id:A084119 - cp's OEIS frontend

A010056 Characteristic function of Fibonacci numbers: a(n) = 1 if n is a Fibonacci number, otherwise 0.

1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Understood as a binary number, Sum_{k>=0} a(k)/2^k, the resulting decimal expansion is 1.910278797207865891... = Fibonacci_binary+0.5 (see A084119) or Fibonacci_binary_constant-0.5 (see A124091), respectively. - Hieronymus Fischer, May 14 2007
a(n)=1 if and only if there is an integer m such that x=n is a root of p(x)=25*x^4-10*m^2*x^2+m^4-16. Also a(n)=1 iff floor(s)<>floor(c) or ceiling(s)<>ceiling(c) where s=arcsinh(sqrt(5)*n/2)/log(phi), c=arccosh(sqrt(5)*n/2)/log(phi) and phi=(1+sqrt(5))/2. - Hieronymus Fischer, May 17 2007
a(A000045(n)) = 1; a(A001690(n)) = 0. - Reinhard Zumkeller, Oct 10 2013
Image, under the map sending a,b,c -> 1, d,e,f -> 0, of the fixed point, starting with a, of the morphism sending a -> ab, b -> c, c -> cd, d -> d, e -> ef, f -> e. - Jeffrey Shallit, May 14 2016

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.
D. Bailey et al., On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux (2004), Volume: 16, Issue: 3, page 487-518.
Wikipedia, Fibonacci number
Index entries for characteristic functions

Cf. A000045, A001690, A072649, A104162, A108852, A124091, A130233, A130234.
Decimal expansion of Fibonacci binary is in A084119.
Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.
Cf. A079586 (Dirich. g.f. at s=1).

import Data.List (genericIndex)
a010056 = genericIndex a010056_list
a010056_list = 1 : 1 : ch [2..] (drop 3 a000045_list) where
   ch (x:xs) fs'@(f:fs) = if x == f then 1 : ch xs fs else 0 : ch xs fs'
-- Reinhard Zumkeller, Oct 10 2013

a:= n-> (t-> `if`(issqr(t+4) or issqr(t-4), 1, 0))(5*n^2):
seq(a(n), n=0..144);  # Alois P. Heinz, Dec 06 2020

Join[{1},With[{fibs=Fibonacci[Range[15]]},If[MemberQ[fibs,#],1,0]& /@Range[100]]]  (* Harvey P. Dale, May 02 2011 *)

a(n)=my(k=n^2);k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)) \\ Charles R Greathouse IV, Jul 30 2012

from sympy.ntheory.primetest import is_square
def A010056(n): return int(is_square(m:=5*n**2-4) or is_square(m+8)) # Chai Wah Wu, Mar 30 2023

G.f.: (Sum_{k>=0} x^A000045(k)) - x. - Hieronymus Fischer, May 17 2007

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

Original entry on oeis.org

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

A124091 Decimal expansion of Fibonacci binary constant: Sum{i>=0} (1/2)^Fibonacci(i).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 25 2006

This constant is transcendental, see A084119. - Charles R Greathouse IV, Nov 12 2014

			2.4102787972078658917940430244710631444834239245952787725932...

Harry J. Smith, Table of n, a(n) for n = 1..20000
D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518.
Index entries for transcendental numbers

Cf. A007404 (Kempner-Mahler number), A125600 (continued fraction), A084119 (essentially the same).

Cf. A000301.

RealDigits[ N[ Sum[(1/2)^Fibonacci[i], {i, 0, Infinity}], 111]][[1]] (* Robert G. Wilson v, Nov 26 2006 *)

a=0 ; for(n=0,30, a += .5^fibonacci(n) ; print(a) ; )

default(realprecision, 20080); x=suminf(k=0, 1/2^fibonacci(k)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b124091.txt", n, " ", d)) \\ Harry J. Smith, May 04 2009

Equals Sum_{i>=0} 1/2^A000045(i).

Equals A084119 + 1.

More terms from Robert G. Wilson v, Nov 26 2006

A006518 Continued fraction for Sum_{k >= 2} 2^(-Fibonacci(k)).

Original entry on oeis.org

0, 1, 10, 6, 1, 6, 2, 14, 4, 124, 2, 1, 2, 2039, 1, 9, 1, 1, 1, 262111, 2, 8, 1, 1, 1, 3, 1, 536870655, 4, 16, 3, 1, 3, 7, 1, 140737488347135, 8, 128, 2, 1, 1, 1, 7, 2, 1, 9, 1

Offset: 0

N. J. A. Sloane

			0.91027879720786589179404302... = 0 + 1/(1 + 1/(10 + 1/(6 + 1/(1 + ...)))). - _Harry J. Smith_, May 04 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..564
V. Meally, Letter to N. J. A. Sloane, May 1975
A. J. van der Poorten, Continued fractions of formal power series
A. J. van der Poorten and Jeffrey Shallit, A specialised continued fraction, Canad. J. Math., 45 (1993), 1067-1079.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A000045, A084119, A124091, A125600, A181313.

{ allocatemem(932245000); default(realprecision, 10000); x=suminf(k=2, 1/2^fibonacci(k)); c=contfrac(x); for (n=1, 565, write("b006518.txt", n-1, " ", c[n])); } \\ Harry J. Smith, May 04 2009

Interestingly, a(13)=2^11-2^3-1, a(19)=2^18-2^5-1, a(27)=2^29-2^8-1, a(35)=2^47-2^13-1. - Ralf Stephan, Jun 07 2005

A181313 Continued fraction expansion of the Fibonacci binary number.

Original entry on oeis.org

1, 2, 2, 3, 2, 28, 1, 13, 1, 2, 1, 123, 1, 6, 1, 2039, 2, 2, 6, 262111, 1, 35, 1, 1, 3, 536870655, 1, 2, 1, 15, 1, 3, 3, 1, 1, 1, 2, 140737488347135, 1, 1, 1, 1, 1, 127, 1, 7, 7, 1, 5, 2, 2, 75557863725914321321983, 1, 1, 2, 5, 1, 2047, 2, 2, 5, 1, 31, 6, 1, 1, 3, 2, 2

Offset: 0

Charles R Greathouse IV, Oct 12 2010

Essentially the same as A125600. - R. J. Mathar, Oct 14 2010

Charles R Greathouse IV, Table of n, a(n) for n = 0..638
D. Bailey, J. Borwein, R. Crandall, and C. Pomerance, On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518.
J. H. Loxton and A. van der Poorten, Arithmetic properties of certain functions in several variables III, Bulletin of the Australian Mathematical Society, Volume 16, Issue 01, February 1977, pp 15-47.
J. Shallit and A. van der Poorten, A specialised continued fraction, Can. J. Math. 45 (1993), 1067-79.
Alf van der Poorten, In thrall to Fibonacci

Cf. A084119 (decimal expansion), A125600 (essentially the same), A006518.

PARI
```
contfrac(suminf(n=1,2.^-fibonacci(n)))
```

Offset changed by Andrew Howroyd, Aug 09 2024

cp's OEIS Frontend

A010056 Characteristic function of Fibonacci numbers: a(n) = 1 if n is a Fibonacci number, otherwise 0.

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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).

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A124091 Decimal expansion of Fibonacci binary constant: Sum{i>=0} (1/2)^Fibonacci(i).

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A006518 Continued fraction for Sum_{k >= 2} 2^(-Fibonacci(k)).

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A181313 Continued fraction expansion of the Fibonacci binary number.

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A121821 Decimal expansion of the Lucas binary number, Sum_{k>0} 1/2^L(k), where L(k) = A000032(k).

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A327558 Decimal expansion of Sum_{k>=1} 1/F(k)! where F(k) = A000045(k).

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