A073115 -id:A073115 - cp's OEIS frontend

A073116 Continued fraction expansion of S/2 where S = Sum_{k>=0} 1/2^floor(k*phi) (A073115) and phi is the golden ratio (1+sqrt(5))/2 (A001622).

0, 1, 5, 1, 8, 4, 64, 128, 16384, 1048576, 34359738368, 18014398509481984, 1237940039285380274899124224, 11150372599265311570767859136324180752990208, 27606985387162255149739023449108101809804435888681546220650096895197184

Offset: 1

Benoit Cloitre, Aug 19 2002

The number S is the number whose digits are obtained from the substitution system (1->(1,0),0->(1)). The n-th term of the continued fraction expansion for S is 2^Fibonacci(n-2) (cf. A000301). This number S is known to be transcendental. The continued fraction of S/2^m follows the same kind of rule as for S/2.

Amiram Eldar, Table of n, a(n) for n = 1..20

Cf. A000045, A000301, A001622, A073115.

a[1] = 0; a[2] = 1; a[3] = 5; a[n_] := 2^(Fibonacci[n - 2] - (-1)^n); Array[a, 15] (* Amiram Eldar, May 08 2022 *)

If n>2, a(2n+1) = 2^(F(2n-1)+1) and a(2n)= 2^(F(2n-2)-1), where F(n) is the n-th Fibonacci number.

More terms from Amiram Eldar, May 08 2022

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

A014565 Decimal expansion of rabbit constant.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Dec 11 1999

Davison shows that the continued fraction is (essentially) A000301 and proves that this constant is transcendental. - Charles R Greathouse IV, Jul 22 2013
Using Davison's result we can find an alternating series representation for the rabbit constant r as r = 1 - sum {n >= 1} (-1)^(n+1)*(1 + 2^Fibonacci(3*n+1))/( (2^(Fibonacci(3*n - 1)) - 1)*(2^(Fibonacci(3*n + 2)) - 1) ). The series converges rapidly: for example, the first 10 terms of the series give a value for r accurate to more than 1.7 million decimal places. See A005614. - Peter Bala, Nov 11 2013
The rabbit constant is the number having the infinite Fibonacci word A005614 as binary expansion; its continued fraction expansion is A000301 = 2^A000045 (after a leading zero, depending on convention). - M. F. Hasler, Nov 10 2018

			0.709803442861291314641787399444575597012502205767...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 439.
M. Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, New York: W. H. Freeman, 1991.

Sean A. Irvine and Joerg Arndt, Table of n, a(n) for n = 0..2000
W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
P. G. Anderson, T. C. Brown, and P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
Joerg Arndt, Matters Computational (The Fxtbook), p. 754.
J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc. 63 (1977), pp. 29-32.
Martin Griffiths, 96.12 The sum of a series: rational or irrational?, The Mathematical Gazette, Vol. 96, No. 535 (2012), pp. 121-124.
Clark Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.
Eric Weisstein's World of Mathematics, Rabbit Constant.
Index entries for transcendental numbers

Cf. A005614, A073115, A119809, A119812.

Take[ RealDigits[ Sum[N[1/2^Floor[k*GoldenRatio], 120], {k, 0, 300}]-1][[1]], 103] (* Jean-François Alcover, Jul 28 2011, after Benoit Cloitre *)
RealDigits[ FromDigits[{Nest[Flatten[# /. {0 -> {1}, 1 -> {1, 0}}] &, {1}, 12], 0}, 2], 10, 111][[1]] (* Robert G. Wilson v, Mar 13 2014 *)
digits = 103; dm = 10; Clear[xi]; xi[b_, m_] := xi[b, m] = RealDigits[ ContinuedFractionK[1, b^Fibonacci[k], {k, 0, m}], 10, digits] // First; xi[2, dm]; xi[2, m = 2 dm]; While[xi[2, m] != xi[2, m - dm], m = m + dm]; xi[2, m] (* Jean-François Alcover, Mar 04 2015, update for versions 7 and up, after advice from Oleg Marichev *)

/* fast divisionless routine from fxtbook */
fa(y, N=17)=
{ my(t, yl, yr, L, R, Lp, Rp);
/* as powerseries correct up to order fib(N+2)-1 */
  L=0; R=1; yl=1; yr=y;
  for(k=1, N, t=yr; yr*=yl; yl=t; Lp=R; Rp=R+yr*L; L=Lp; R=Rp; );
  return( R )
}
a=0.5*fa(0.5) /* Joerg Arndt, Apr 15 2010 */

my(r=1,p=(3-sqrt(5))/2,n=1);while(r>r-=1.>>(n\p),n++);A014565=r \\ M. F. Hasler, Nov 10 2018

my(f(n)=1.<A098317 (=> 298, 1259, 5331, ... digits). - M. F. Hasler, Nov 10 2018

Equals Sum_{n>=1} 1/2^b(n) where b(n) = floor(n*phi) = A000201(n).
Equals -1 + A073115.
From Peter Bala, Nov 04 2013: (Start)
The results of Adams and Davison 1977 can be used to find a variety of alternative series representations for the rabbit constant r. Here are several examples (phi denotes the golden ratio (1/2)*(1 + sqrt(5))).
r = Sum_{n >= 2} ( floor((n+1)*phi) - floor(n*phi) )/2^n = (1/2)*Sum_{n >= 1} A014675(n)/2^n.
r = Sum_{n >= 1} floor(n/phi)/2^n = Sum_{n >= 1} A005206(n-1)/2^n.
r = ( Sum_{n >= 1} 1/2^floor(n/phi) ) - 2 and r = ( Sum_{n >= 1} floor(n*phi)/2^n ) - 2 = ( Sum_{n >= 1} A000201(n)/2^n ) - 2.
More generally, for integer N >= -1, r = ( Sum_{n >= 1} 1/2^floor(n/(phi + N)) ) - (2*N + 2) and for all integer N, r = ( Sum_{n >= 1} floor(n*(phi + N))/2^n ) - (2*N + 2).
Also r = 1 - Sum_{n >= 1} 1/2^floor(n*phi^2) = 1 - Sum_{n >= 1} 1/2^A001950(n) and r = 1 - Sum_{n >= 1} floor(n*(2 - phi))/2^n = 1 - Sum_{n >= 1} A060144(n)/2^n. (End)

More terms from Simon Plouffe, Dec 11 1999

A119812 Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} A049472(n)/2^n = Sum_{n>=1} 1/2^A001951(n).

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, May 26 2006

Dual constant: A119809 = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>=1} A001951(n)/2^n. The binary expansion of this constant is given by A080764 with offset n=1. Plouffe's Inverter describes approximations to this constant as "polylogarithms type of series with the floor function [ ]."

			c = 0.858267656461002055792260308433375148664905190083506778667684867..
Continued fraction (A119813):
c = [0;1,6,18,1032,16777344,288230376151842816,...]
where partial quotients are given by:
PQ[n] = 4^A000129(n-2) + 2^A001333(n-3) (n>2), with PQ[1]=0, PQ[2]=1.
The following are equivalent expressions for the constant:
(1) Sum_{n>=1} A049472(n)/2^n; A049472(n)=[n/sqrt(2)];
(2) Sum_{n>=1} 1/2^A001951(n); A001951(n)=[n*sqrt(2)];
(3) Sum_{n>=1} A080764(n)/2^n; A080764(n)=[(n+1)/sqrt(2)]-[n/sqrt(2)];
where [x] = floor(x).
These series illustrate the above expressions:
(1) c = 0/2^1 + 1/2^2 + 2/2^3 + 2/2^4 + 3/2^5 + 4/2^6 + 4/2^7 +...
(2) c = 1/2^1 + 1/2^2 + 1/2^4 + 1/2^5 + 1/2^7 + 1/2^8 + 1/2^9 +...
(3) c = 1/2^1 + 1/2^2 + 0/2^3 + 1/2^4 + 1/2^5 + 0/2^6 + 1/2^7 +...

W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
P. G. Anderson, T. C. Brown, P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity Proc. Amer. Math. Soc. 123 (1995), 2005-2009.

Cf. A119813 (continued fraction), A119814 (convergents); A119809 (dual constant); A000129 (Pell), A001333; Beatty sequences: A049472, A001951, A080764; variants: A014565 (rabbit constant), A073115.

{a(n)=local(t=sqrt(2)/2,x=sum(m=1,10*n,floor(m*t)/2^m));floor(10^n*x)%10}

Removed leading zero and corrected offset R. J. Mathar, Feb 05 2009

A119809 Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>=1} A001951(n)/2^n.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, May 26 2006

Dual constant: A119812 = Sum_{n>=1} A049472(n)/2^n = Sum_{n>=1} 1/2^A001951(n). Since this constant c = 2 + Sum_{n>=1} 1/2^A003151(n), where A003151(n) = n + floor(n*sqrt(2)), then the binary expansion of the fractional part of c has 1's only at positions given by Beatty sequence A003151(n) and zeros elsewhere. Plouffe's Inverter describes approximations to the fractional part of c as "polylogarithms type of series with the floor function [ ]."

			c = 2.32258852258806773012144068278798408011950250800432925665718...
Continued fraction (A119810):
c = [2;3,10,132,131104,2199023259648,633825300114114700748888473600,..]
where partial quotients are given by:
PQ(n) = 2^A001333(n-1) + 2^A000129(n-2) (n>1), with PQ(1)=2.
The following are equivalent expressions for the constant:
(1) Sum_{n>=1} 1/2^A049472(n); A049472(n)=[n/sqrt(2)];
(2) Sum_{n>=1} A001951(n)/2^n; A001951(n)=[n*sqrt(2)];
(3) Sum_{n>=1} 1/2^A003151(n) + 2; A003151(n)=[n*sqrt(2)]+n;
(4) Sum_{n>=1} 1/2^A097508(n) - 2; A097508(n)=[n*sqrt(2)]-n;
(5) Sum_{n>=1} A006337(n)/2^n + 1; A006337(n)=[(n+1)*sqrt(2)]-[n*sqrt(2)];
where [x] = floor(x).
These series illustrate the above expressions:
(1) c = 1/2^0 + 1/2^1 + 1/2^2 + 1/2^2 + 1/2^3 + 1/2^4 + 1/2^4 +...
(2) c = 1/2^1 + 2/2^2 + 4/2^3 + 5/2^4 + 7/2^5 + 8/2^6 + 9/2^7 +...
(3) c = 2 + 1/2^2 + 1/2^4 + 1/2^7 + 1/2^9 + 1/2^12 + 1/2^14 +...
(4) c =-2 + 1/2^0 + 1/2^0 + 1/2^1 + 1/2^1 + 1/2^2 + 1/2^2 + 1/2^2 +...
(5) c = 1 + 1/2^1 + 2/2^2 + 1/2^3 + 2/2^4 + 1/2^5 + 1/2^6 + 2/2^7 +...

Kevin O'Bryant, A generating function technique for Beatty sequences and other step sequences, Journal of Number Theory, Volume 94, Issue 2, June 2002, Pages 299-319.

Cf. A119810 (continued fraction), A119811 (convergents); A119812 (dual constant); A000129 (Pell), A001333; Beatty sequences: A049472, A001951, A003151, A097508, A006337; variants: A014565 (rabbit constant), A073115.

Cf. A081544, A081550, A081573.

{a(n)=local(t=sqrt(2),x=sum(m=1,10*n,floor(m*t)/2^m));floor(10^n*x)%10}

Equals Sum(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < sqrt(2) (O'Bryant, 2002). - Amiram Eldar, May 25 2023

A081544 Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < phi, where phi is the Golden ratio.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Apr 21 2003

Kevin O'Bryant, A generating function technique for Beatty sequences and other step sequences, Journal of Number Theory, Volume 94, Issue 2, June 2002, Pages 299-319.

Cf. A001622 (golden ratio), A014565, A073115.

Cf. A081550, A081564, A081573.

With[{digmax = 120}, RealDigits[Sum[1/2^Floor[k/GoldenRatio], {k, 1, 10*digmax}], 10, digmax][[1]]] (* Amiram Eldar, May 25 2023 *)

Equals Sum_{k>=1} (1/2)^floor(k/phi).

Equals A014565 + 2 = A073115 + 1. - Amiram Eldar, May 25 2023

Data corrected by Amiram Eldar, May 25 2023

cp's OEIS Frontend

A073116 Continued fraction expansion of S/2 where S = Sum_{k>=0} 1/2^floor(k*phi) (A073115) and phi is the golden ratio (1+sqrt(5))/2 (A001622).

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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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A014565 Decimal expansion of rabbit constant.

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A119812 Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} A049472(n)/2^n = Sum_{n>=1} 1/2^A001951(n).

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A119809 Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>=1} A001951(n)/2^n.

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A081544 Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < phi, where phi is the Golden ratio.

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