A052119 -id:A052119 - cp's OEIS frontend

A222466 Decimal expansion of the limit of the continued fraction 1/(1+2/(2+2/(3+2/(4+... in terms of Bessel functions.

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

Offset: 0

Wolfdieter Lang, Mar 07 2013

The continued fraction (0 + K_{k=1..oo} (2/k))/2 = 1/(1+2/(2+2/(3+2/(4+ ... converges, and its limit is given in the formula section in terms of Bessel functions.
In general, the continued fraction 0 + K_{k=1..oo} (x/k) = x/(1+x/(2+x/(3+... has n-th approximation x*Phat(n,x)/ Q(n,x), with the row polynomials Phat of A221913 and Q of A084950. These polynomials are written in terms of Bessel functions. Divided by n! = Gamma(n+1) one knows the limit for n -> infinity for these two polynomial systems. This results in the given formula 0 + K_{k=1..oo} (x/k) = sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x)).
For x=1 see for the limit of the continued fraction A052119 and for the n-th approximation A001053(n+1)/A001040(n+1).

			0.5631786198117113854257529037035635...

G. C. Greubel, Table of n, a(n) for n = 0..5000

A052119 (x=1), 2*A221913/A084950.

RealDigits[BesselI[1, 2*Sqrt[2]]/(Sqrt[2]*BesselI[0, 2*Sqrt[2]]), 10, 50][[1]] (* G. C. Greubel, Aug 16 2017 *)

default(realprecision, 120);
sqrt(2)*besseli(1,2*sqrt(2))/(2*besseli(0,2*sqrt(2))) \\ Rick L. Shepherd, Jan 18 2014

(0 + K_{k=1..oo} (2/k))/2 = 1/(1+2/(2+2/(3+2/(4+ ... =

sqrt(2)*BesselI(1,2*sqrt(2))/(2*BesselI(0,2*sqrt(2)))

Offset corrected and terms added by Rick L. Shepherd, Jan 18 2014

A309930 Decimal expansion of the constant whose continued fraction representation is the cubes [0; 1, 8, 27, 64, ...], A000578.

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Nov 11 2019

			0.8893440000327626936054947063212219810354292088...

Cf. A000578, A052119, A073824.

N[FromContinuedFraction[Table[k^3, {k, 0, 1000}]], 120] (* Vaclav Kotesovec, Nov 20 2019 *)

dec_exp(v)= w=contfracpnqn(v); w[1, 1]/w[2, 1]+0.
dec_exp(vector(2000, i, (i-1)^3)) \\ Michel Marcus, Nov 19 2019; after A073824

import decimal
from decimal import Decimal as D
def constant_from_cofr(clist):
    hn0, kn0 = 0, 1
    hn1, kn1 = 1, 0
    for n in clist:
        hn2 = (n * hn1) + hn0
        kn2 = (n * kn1) + kn0
        hn0, kn0 = hn1, kn1
        hn1, kn1 = hn2, kn2
    return D(hn2)/D(kn2)
if _name_ == "_main_":
    prec = 200
    decimal.getcontext().prec = prec
    glist = [x**3 for x in range(500)]
    print(', '.join(str(x) for x in str(constant_from_cofr(glist))[2:]))

A214070 Decimal expansion of the number whose continued fraction is 1, 2, 4, 8, 16, ...

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 06 2012

Essentially the same as A096641. - R. J. Mathar, Aug 10 2012

			1.4459346405122026681195543406826176842704088452034385079032635605006619006916...

Cf. A000079, A096641, A275614.

Cf. A052119, A073824, A309930.

RealDigits[ FromContinuedFraction[{1, 2^Range@ 19}], 10, 111][[1]]

From Amiram Eldar, Feb 08 2022: (Start)
Equals A096641 + 1.
Equals 1/A275614. (End)

A222471 Decimal expansion of the negative of the limit of the continued fraction 1/(1-2/(2-2/(3-2/(4-... in terms of Bessel functions.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Mar 23 2013

The continued fraction (0 + K_{k=1..oo} (-2/k))/(-2) = 1/(1-2/(2-2/(3-2/(4- ... converges, and its negative limit is given in the formula section in terms of Bessel functions.
In general, the continued fraction 0 + K_{k=1..oo} (x/k) = x/(1+x/(2+x/(3+... has n-th approximation x*Phat(n,x)/Q(n,x), with the row polynomials Phat of A221913 and Q of A084950. These polynomials are written in terms of Bessel function. Divided by n! = Gamma(n+1) one knows the limit for n -> infinity for these two polynomial systems for given x. This results in the formula 0 + K_{k=1..oo} (x/k) = sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x)).
For x=1 see for the limit of the continued fraction A052119 and for the n-th approximation A001053(n+1)/A001040(n+1).

			-1.4397493218702328058...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A052119 (x=1), A222466 (x=2), A222469/A222470.

RealDigits[BesselJ[1, 2*Sqrt[2]]/(Sqrt[2]*BesselJ[0, 2*Sqrt[2]]), 10, 50][[1]] (* G. C. Greubel, Aug 16 2017 *)

besselj(1,sqrt(8))/besselj(0,sqrt(8))/sqrt(2) \\ Charles R Greathouse IV, Feb 19 2014

Equals (1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/BesselJ(0,2*sqrt(2)).

A247844 Decimal expansion of the value of the continued fraction [1; 1, 2, 3, 4, 5, ...].

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 25 2014

Equals 1+A052119.

			1.697774657964007982006790592551752599486658262998...

MathOverflow, Is any particular algebraic number known to have unbounded continued fraction coefficients?
Eric Weisstein's MathWorld, Continued Fraction Constant
Eric Weisstein's MathWorld, Continued Fraction

Cf. A001040, A001053, A052119, A060997, A070910, A096789.

FromContinuedFraction[Join[{1}, Range[50]]] // RealDigits[#, 10, 105]& // First
(* or *) 1+BesselI[1, 2]/BesselI[0, 2] // RealDigits[#, 10, 105]& // First

1+besseli(1,2)/besseli(0,2) \\ Charles R Greathouse IV, Oct 23 2023

1 + I_1(2) / I_0(2), where I_n(x) gives the modified Bessel function of the first kind.

A347052 Decimal expansion of the continued fraction 1/(12 + 1/(23 + 1/(34 + 1/(45 + 1/(5*6 + ...))))).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 13 2021

			0.46202332508023861850355914941757191597703...

Cf. A002378, A052119, A073823, A073824, A346960, A347051.

terms=106; RealDigits[ContinuedFractionK[k(k+1),{k,terms}],10,terms][[1]] (* Stefano Spezia, Aug 23 2025 *)

Equals lim_{n -> oo} A346960(n)/A347051(n).

A266579 Numerator of the continued fraction [Sum_{k=0..n} k; 1, 2, 3,..., n].

Original entry on oeis.org

2, 11, 67, 460, 3532, 30225, 286289, 2979896, 33852226, 417123475, 5543942107, 79086006756, 1205573749892, 19561113090785, 336643494142657, 6125614986385360, 117514626855080914, 2370682022353448571, 50173196512398036851, 1111614380526424428380

Offset: 1

Ilya Gutkovskiy, May 07 2016

			2, 11/3, 67/10, 460/43, 3532/225, 30225/1393, 286289/9976, 2979896/81201, 33852226/740785, 417123475/7489051, 5543942107/83120346, 79086006756/1004933203, 1205573749892/13147251985, 19561113090785/185066460993,...
a(10) = 417123475 because 55+1/(1+1/(2+1/(3+1/(4+1/(5+1/(6+1/(7+1/(8+1/(9+1/10))))))))) = 417123475/7489051 and 1+2+3+4+5+6+7+8+9+10 = 55.

Ilya Gutkovskiy, Extended example
Eric Weisstein's World of Mathematics, Continued Fraction Constant
Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A001040 (denominator, offset 2), A001053, A052119.

Table[Numerator[n ((n + 1)/2) + ContinuedFractionK[1, k, {k, n}]], {n, 20}]

a(n) = A001040(n+1)*A000217(n) + A001053(n+1).

A293577 Decimal expansion of number with continued fraction expansion 0, 1, 12, 123, 1234, 12345, 123456, ... (A007908).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Oct 12 2017

			0.92312499968345024117401233060984219166367488... = 1/(1 + 1/(12 + 1/(123 + 1/(1234 + 1/(12345 + 1/(123456 + 1/...)))))).

G. C. Greubel, Table of n, a(n) for n = 0..5000 [a(5000) corrected by _Georg Fischer_, Apr 02 2025]

Cf. A007908, A030167, A052119, A264934.

Take[RealDigits[N[FromContinuedFraction[Table[FromDigits[Flatten[IntegerDigits[Range[n]]]], {n, 0, 20}]], 101]][[1]], 100]  (* modified by Ilya Gutkovskiy, Nov 08 2017 *)

a(99) corrected by G. C. Greubel, Nov 07 2017

A296140 Decimal expansion of 1/sqrt(1 + 1/sqrt(2 + 1/sqrt(3 + 1/sqrt(4 + 1/sqrt(5 + ...))))).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 05 2017

			0.7837663092363964699519430776387428127041411807738774755...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Nested Radical Constant

Cf. A052119, A060006, A072449.

RealDigits[Fold[1/Sqrt[#1 + #2]&, 0, Range[100, 1, -1]], 10, 100][[1]] (* Jean-François Alcover, Dec 19 2017 *)

A365052 Decimal expansion of continued fraction [1; 4, 9, 16, 25, ... n^2, ... ].

Original entry on oeis.org

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

Offset: 1

Rok Cestnik, Aug 18 2023

			1.243288478399715644...

Cf. A073824 (reciprocal), A036246/A036245 (convergents).

Cf. A226771, A060997, A052119.

A365052 = RealDigits[FromContinuedFraction[Range[1,50]^2],10,#][[1]]&;

p(N) = my(m=contfracpnqn(vector(N, i, i^2))); m[1,1]/m[2,1];
A365052(N) = {my(t=2); while(floor(10^N*p(t)) != floor(10^N*p(t+1)), t++); digits(floor(10^(N-1)*p(t)))};

Equals 1/A073824.

cp's OEIS Frontend

A222466 Decimal expansion of the limit of the continued fraction 1/(1+2/(2+2/(3+2/(4+... in terms of Bessel functions.

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A309930 Decimal expansion of the constant whose continued fraction representation is the cubes [0; 1, 8, 27, 64, ...], A000578.

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A214070 Decimal expansion of the number whose continued fraction is 1, 2, 4, 8, 16, ...

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A222471 Decimal expansion of the negative of the limit of the continued fraction 1/(1-2/(2-2/(3-2/(4-... in terms of Bessel functions.

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A247844 Decimal expansion of the value of the continued fraction [1; 1, 2, 3, 4, 5, ...].

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A347052 Decimal expansion of the continued fraction 1/(1*2 + 1/(2*3 + 1/(3*4 + 1/(4*5 + 1/(5*6 + ...))))).

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A266579 Numerator of the continued fraction [Sum_{k=0..n} k; 1, 2, 3,..., n].

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A293577 Decimal expansion of number with continued fraction expansion 0, 1, 12, 123, 1234, 12345, 123456, ... (A007908).

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A347052 Decimal expansion of the continued fraction 1/(12 + 1/(23 + 1/(34 + 1/(45 + 1/(5*6 + ...))))).