A124091 -id:A124091 - cp's OEIS frontend

A125600 Continued fraction expansion of constant defined in A124091.

2, 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

Robert G. Wilson v, Nov 26 2006

Progressively larger PQ's: 2, 3, 28, 123, 2039, 262111, 536870655, 140737488347135, 75557863725914321321983, 10633823966279326983230456465062887423, 803469022129495137770981046170581301261101460862599398686719, 8543948143683640329580086824678208458410818089426611079788166431288878284152542557401710898184191, ...,.

Cf. A124091 (decimal expansion), A181313 (essentially the same), A006518.

c = N[Sum[(1/2)^Fibonacci[i], {i, 0, Infinity}], 1000]; ContinuedFraction@c

Offset changed by Andrew Howroyd, Aug 03 2024

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

Original entry on oeis.org

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

A000301 a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n).

Original entry on oeis.org

1, 2, 2, 4, 8, 32, 256, 8192, 2097152, 17179869184, 36028797018963968, 618970019642690137449562112, 22300745198530623141535718272648361505980416, 13803492693581127574869511724554050904902217944340773110325048447598592

Offset: 0

N. J. A. Sloane, Mar 15 1996

Continued fraction expansion of s = A073115 = 1.709803442861291... = Sum_{k >= 0} (1/2^floor(k * phi)) where phi is the golden ratio (1 + sqrt(5))/2. - Benoit Cloitre, Aug 19 2002

The continued fraction expansion of the above constant s is [1; 1, 2, 2, 4, ...], that of the rabbit constant r = s-1 = A014565 is [0; 1, 2, 2, 4, ...]. - M. F. Hasler, Nov 10 2018

Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 913.

T. D. Noe, Table of n, a(n) for n = 0..18
Manosij Ghosh Dastidar and Michael Wallner, Bijections between Variants of Dyck Paths and Integer Compositions, arXiv:2406.16404 [math.CO], 2024. See p. 1.
J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc., 63 (1977), 29-32.
Samuele Giraudo, Intervals of balanced binary trees in the Tamari lattice, arXiv preprint arXiv:1107.3472 [math.CO], 2011-2012, and Theor Comput Sci 420 (2012) 1-27.
Bertrand Teguia Tabuguia, Computing with D-Algebraic Sequences, arXiv:2412.20630 [math.AG], 2024. See p. 9.
Index to divisibility sequences

Cf. A000045, A000079, A000304, A010098, A010099, A010100, A073115, A124091.

Column k = 2 of A244003.

a000301 = a000079 . a000045
a000301_list = 1 : scanl (*) 2 a000301_list
-- Reinhard Zumkeller, Mar 20 2013

[2^Fibonacci(n): n in [0..20]]; // Vincenzo Librandi, Apr 18 2011

A000301 := proc(n) option remember;
             if n < 2 then 1+n
           else A000301(n-1)*A000301(n-2)
             fi
           end:
seq(A000301(n), n=0..15);

2^Fibonacci[Range[0, 14]] (* Alonso del Arte, Jul 28 2016 *)

a(n)=1<Charles R Greathouse IV, Jan 12 2012

[2^fibonacci(n) for n in range(15)] # G. C. Greubel, Jul 29 2024

a(n) ~ k^phi^n with k = 2^(1/sqrt(5)) = 1.3634044... and phi the golden ratio. - Charles R Greathouse IV, Jan 12 2012
a(n) = A000304(n+3) / A010098(n+1). - Reinhard Zumkeller, Jul 06 2014
Sum_{n>=0} 1/a(n) = A124091. - Amiram Eldar, Oct 27 2020
Limit_{n->oo} a(n)/a(n-1)^phi = 1. - Peter Woodward, Nov 24 2023

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 18 2011

A084119 Decimal expansion of the Fibonacci binary number, Sum_{k>0} 1/2^F(k), where F(k) = A000045(k).

Original entry on oeis.org

1, 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

Offset: 1

Ralf Stephan, May 18 2003

The Fibonacci binary number 1.41027879720... is known to be transcendental.

			1.410278797207865891794043024471063...

Harry J. Smith, Table of n, a(n) for n = 1..20000
David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Carl 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.
A. J. Van Der Poorten and J. Shallit, A specialised continued fraction, Can. J. Math. 45 (1993), 1067-79.
Index entries for transcendental numbers.

Cf. A000045, A010056, A079586, A181313 (continued fraction), A124091 (essentially the same).

RealDigits[N[Sum[1/2^Fibonacci[k], {k, 1, Infinity}], 120]][[1]] (* Amiram Eldar, Jun 12 2023 *)

suminf(k=1,1/2^fibonacci(k)) \\ This gives the Fibonacci binary number, not the sequence

default(realprecision, 20080); x=suminf(k=1, 1/2^fibonacci(k)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b084119.txt", n, " ", d)); \\

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

A309537 Total number of Fibonacci parts in all compositions of n.

Original entry on oeis.org

0, 1, 3, 8, 19, 46, 106, 241, 541, 1198, 2629, 5724, 12380, 26625, 56978, 121413, 257740, 545308, 1150272, 2419856, 5078336, 10633921, 22222338, 46353669, 96525324, 200686620, 416645184, 863834256, 1788756288, 3699688128, 7643727360, 15776156928, 32529718272

Offset: 0

Alois P. Heinz, Aug 06 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..3312

Cf. A000045, A102291, A124091, A144115.

a:= proc(n) option remember; add(a(n-j)+`if`((t->issqr(t+4)
      or issqr(t-4))(5*j^2), ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);

a[n_] := a[n] = Sum[a[n - j] + With[{t = 5 j^2}, If[IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4], Ceiling[2^(n - j - 1)], 0]], {j, 1, n}];
a /@ Range[0, 33] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

G.f.: Sum_{k>=2} x^Fibonacci(k)*(1-x)^2/(1-2*x)^2.
a(n) ~ c * 2^n * n, where c = 0.22756969930196647294851075611776578612085598114... - Vaclav Kotesovec, Aug 18 2019
c = A124091/4 - 3/8. - Vaclav Kotesovec, Mar 17 2024

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A125600 Continued fraction expansion of constant defined in A124091.

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

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A000301 a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n).

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A084119 Decimal expansion of the Fibonacci binary number, Sum_{k>0} 1/2^F(k), where F(k) = A000045(k).

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

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A309537 Total number of Fibonacci parts in all compositions of n.

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