A119809 -id:A119809 - cp's OEIS frontend

A119811 Numerators of the convergents to the continued fraction for the constant A119809 defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>=1} A001951(n)/2^n.

2, 7, 72, 9511, 1246930216, 2742028548141904733479, 1737967067447512977484869808775151193351704374584616

Offset: 1

Paul D. Hanna, May 26 2006

The number of digits in these numerators are (beginning at n=1): [1,1,2,4,10,22,52,124,297,717,1729,4173,10074,24319,58709,141735,..].

			c = 2.32258852258806773012144068278798408011950250800432925665718...
Convergents begin:
[2/1, 7/3, 72/31, 9511/4095, 1246930216/536870911,...]
where the denominators of the convergents equal [2^A000129(n-1)-1]:
[1,3,31,4095,536870911,1180591620717411303423,...],
and A000129 is the Pell numbers.

Cf. A119809 (constant), A119811 (continued fraction), A000129; A119812 (dual constant).

{a(n)=local(M=contfracpnqn(vector(n,k,if(k==1,2, 2^round(((1+sqrt(2))^(k-1)+(1-sqrt(2))^(k-1))/2) +2^round(((1+sqrt(2))^(k-2)-(1-sqrt(2))^(k-2))/(2*sqrt(2))))))); return(M[1,1])}

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

A119813 Partial quotients of the continued fraction of the constant A119812 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

0, 1, 6, 18, 1032, 16777344, 288230376151842816, 1393796574908163946345982392042721617379328

Offset: 1

Paul D. Hanna, May 26 2006

Convergents begin: [0/1, 1/1, 6/7, 109/127, 112494/131071,...], where the denominators of the convergents are equal to [2^A001333(n-1)-1], where A001333 are numerators of continued fraction convergents to sqrt(2). The number of digits in these partial quotients are (beginning at n=2): [1,1,2,4,8,18,43,102,246,594,1432,3457,8345,20146,48636,117417,...].

			c = 0.858267656461002055792260308433375148664905190083506778667684867..
The partial quotients start:
a(1) = 0; a(2) = 1; a(3) = 4^1 + 2^1; a(4) = 4^2 + 2^1;
a(5) = 4^5 + 2^3; a(6) = 4^12 + 2^7; a(7) = 4^29 + 2^17;
and continue as a(n) = 4^A000129(n-2) + 2^A001333(n-3) where
A000129(n) = ( (1+sqrt(2))^n - (1-sqrt(2))^n )/(2*sqrt(2));
A001333(n) = ( (1+sqrt(2))^n + (1-sqrt(2))^n )/2.

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. A119812 (constant), A119814 (convergents); A119809 (dual constant).

{a(n)=if(n==1,0,if(n==2,1, 4^round(((1+sqrt(2))^(n-2)+(1-sqrt(2))^(n-2))/(2*sqrt(2))) +if(n==3,2,2^round(((1+sqrt(2))^(n-3)-(1-sqrt(2))^(n-3))/2))))}

a(n) = 4^A000129(n-2) + 2^A001333(n-3) for n>2, with a(1)=0, a(2)=1.

A119814 Numerators of the convergents to the continued fraction for the constant A119812 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

0, 1, 6, 109, 112494, 1887350536045, 543991754934632523092182415214, 758213844806172103575972149363453352380811718063209070444420739586832237

Offset: 1

Paul D. Hanna, May 26 2006

The number of digits in these numerators are (beginning at n=2): [1,1,3,6,13,30,72,174,420,1013,2444,5901,14245,34391,83027,...].

			c = 0.858267656461002055792260308433375148664905190083506778667684867..
Convergents begin:
[0/1, 1/1, 6/7, 109/127, 112494/131071, 1887350536045/2199023255551,..]
where the denominators of the convergents equal [2^A001333(n-1)-1]:
[1,1,7,127,131071,2199023255551,633825300114114700748351602687,...]
and A001333 is numerators of continued fraction convergents to sqrt(2).

Cf. A119812 (constant), A119813 (continued fraction), A001333; A119809 (dual constant).

{a(n)=local(M=contfracpnqn(vector(n,k,if(k==1,0,if(k==2,1, 4^round(((1+sqrt(2))^(k-2)+(1-sqrt(2))^(k-2))/(2*sqrt(2))) +if(k==3,2,2^round(((1+sqrt(2))^(k-3)-(1-sqrt(2))^(k-3))/2))))))); return(M[1,1])}

A119810 Partial quotients of the continued fraction 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, 10, 132, 131104, 2199023259648, 633825300114114700748888473600, 883423532389192164791648750371459257913741948437810659652423818057613312

Offset: 1

Paul D. Hanna, May 26 2006

Convergents A119811: [2/1,7/3,72/31,9511/4095,1246930216/536870911,...], where the denominators of the convergents are equal to [2^A000129(n-1)-1] and A000129 is the Pell numbers. The number of digits in these partial quotients are (beginning at n=1): [1,1,2,3,6,13,30,72,174,420,1013,2445,5901,14246,34391,83027,...].

			c = 2.32258852258806773012144068278798408011950250800432925665718...
The partial quotients start:
a(1) = 2^1; a(2) = 2^1 + 2^0; a(3) = 2^3 + 2^1;
a(4) = 2^7 + 2^2; a(5) = 2^17 + 2^5; a(6) = 2^41 + 2^12;
and continue as a(n) = 2^A001333(n-1) + 2^A000129(n-2) where
A001333(n) = ( (1+sqrt(2))^n + (1-sqrt(2))^n )/2;
A000129(n) = ( (1+sqrt(2))^n - (1-sqrt(2))^n )/(2*sqrt(2)).

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.
J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc., 63 (1977), 29-32.

Cf. A119809 (decimal expansion), A119811 (convergents); A119812 (dual constant).

(* b = A001333 *) b[0] = 1; b[1] = 1; b[n_] := b[n] = 2 b[n-1] + b[n-2]; a[1] = 2; a[n_] := 2^b[n-1] + 2^Fibonacci[n-2, 2]; Array[a, 10] (* Jean-François Alcover, May 04 2017 *)

{a(n)=if(n==1,2,2^round(((1+sqrt(2))^(n-1)+(1-sqrt(2))^(n-1))/2) +2^round(((1+sqrt(2))^(n-2)-(1-sqrt(2))^(n-2))/(2*sqrt(2))))}

a(n) = 2^A001333(n-1) + 2^A000129(n-2) for n>1, with a(1)=2.

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A081564 Erroneous duplicate of A119809.

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A119811 Numerators of the convergents to the continued fraction for the constant A119809 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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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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A119813 Partial quotients of the continued fraction of the constant A119812 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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A119814 Numerators of the convergents to the continued fraction for the constant A119812 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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A119810 Partial quotients of the continued fraction 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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