A073824 -id:A073824 - cp's OEIS frontend

A036245 Numerator of fraction equal to the continued fraction [ 0, 1, 4, ..., n^2 ].

1, 4, 37, 596, 14937, 538328, 26393009, 1689690904, 136891356233, 13690825314204, 1656726754374917, 238582343455302252, 40322072770700455505, 7903364845400744581232, 1778297412287938231232705, 455252040910557587940153712, 131569618120563430852935655473

Offset: 1

Jeff Burch

Seiichi Manyama, Table of n, a(n) for n = 1..253

Cf. A036246, A073824.

Table[Numerator[FromContinuedFraction[Range[0, n]^2]], {n, 1, 20}] (* Vaclav Kotesovec, Aug 14 2021 *)

A036245(n) = my(v=vector(n+1)); for(i=1, n+1, if(i==1, v[i]=0, if(i==2, v[i]=1, v[i]=(i-1)^2*v[i-1]+v[i-2]))); v[n+1] \\ Jianing Song, Nov 30 2019

a(n) = n^2 * a(n-1) + a(n-2) for n > 2. - Seiichi Manyama, Jun 05 2018
Lim_{n->oo} a(n)/A036246(n) = A073824. - Jianing Song, Nov 30 2019
a(n) ~  c * n^(2*n + 1) / exp(2*n), where c = 6.5347337470474831902516177263695578212049901774805425962967688345920604685... - Vaclav Kotesovec, Aug 14 2021

More terms from Seiichi Manyama, Jun 05 2018

A036246 CONTINUANT transform of squares 1, 4, 9, ...

Original entry on oeis.org

1, 5, 46, 741, 18571, 669297, 32814124, 2100773233, 170195445997, 17021645372933, 2059789285570890, 296626678767581093, 50131968501006775607, 9826162452876095600065, 2210936683865622516790232, 566009617232052240393899457, 163578990316746963096353733305

Offset: 1

Jeff Burch

Denominator of fraction equal to the continued fraction [ 0, 1, 4, ...n^2 ].

Alois P. Heinz, Table of n, a(n) for n = 1..100
N. J. A. Sloane, Transforms

Cf. A036245, A073824.

a:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, n^2 *a(n-1) +a(n-2)))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 06 2013

Table[Denominator[FromContinuedFraction[Range[0,n]^2]],{n,20}] (* Harvey P. Dale, Jul 16 2017 *)

A036246(n) = my(v=vector(n+1)); for(i=1, n+1, if(i==1, v[i]=1, if(i==2, v[i]=1, v[i]=(i-1)^2*v[i-1]+v[i-2]))); v[n+1] \\ Jianing Song, Nov 30 2019

a(n) ~  c * n^(2*n + 1) / exp(2*n), where c = 8.1245591771139376779472290412302409841950717664641832772206241208918274428499... - Vaclav Kotesovec, Jun 05 2018
From Jianing Song, Nov 30 2019: (Start)
a(n) = n^2 * a(n-1) + a(n-2) for n > 2.
Lim_{n->oo} A036245(n)/a(n) = A073824. (End)

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)

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).

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.

A261827 Decimal expansion of the number whose continued fraction expansion consists of the perfect numbers (A000396).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Sep 02 2015

			6.0357117143069233346283990529260946180806175748136895461...

Eric Weisstein's World of Mathematics, Continued Fraction
Eric Weisstein's World of Mathematics, Perfect Number

Cf. A000396, A073821, A073822, A073823, A073824, A052119, A064442.

ind = {1, 2, 3, 4, 6, 7, 8, 11, 18, 24, 28, 31, 98, 111} (* from A016027 *); p = Prime@ ind; pn = (2^p - 1)(2^(p - 1)); RealDigits[ FromContinuedFraction@ pn, 10, 111][[1]] (* Robert G. Wilson v, Sep 13 2015 *)

A329810 Decimal expansion of the constant whose continued fraction representation is [0; 1, 3, 7, 15, 31, ...] = A000225 (the Mersenne numbers).

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Nov 21 2019

Since Mersenne numbers of the form 2^x - 1 consist entirely of 1's when written in binary, this continued fraction is nothing but 1's if written in binary.

Binary continued fraction: 1/(1+1/(11+1/(111+1/(1111+1/(11111+1/(111111+1/...

			0.758542308171055739268126048842248893421247779...

Cf. A000225, A052119, A073824.

N[FromContinuedFraction[Table[2^k - 1, {k, 0, 100}]], 120] (* Vaclav Kotesovec, Nov 21 2019 *)

dec_exp(v)= w=contfracpnqn(v); w[1, 1]/w[2, 1]+0.
dec_exp(vector(200, i, 2^(i-1)-1)) \\ Michel Marcus, Nov 21 2019

A330156 Decimal expansion of the continued fraction expansion [1; 1/2, 1/3, 1/4, 1/5, 1/6, ...].

Original entry on oeis.org

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

Offset: 1

Daniel Hoyt, Dec 03 2019

This constant is formed from the continued fraction [1; 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ...] the reciprocals of the positive integers, A000027.

			1.7519383938841086612039097015114538792503980068057415636404709501399828870437...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.4, p. 23.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 562.
Wikipedia, Continued Fraction Expansions of Pi.
Index entries for transcendental numbers.

Cf. A000027, A000796, A052119, A073824, A309091.

First[RealDigits[2/(Pi - 2), 10, 100]] (* Paolo Xausa, Apr 27 2024 *)

2 / (Pi - 2) \\ Michel Marcus, Dec 05 2019

1/atan(cotan(1)) \\ Daniel Hoyt, Apr 11 2020

Equals 2 / (Pi - 2).
Equals 1/arctan(cot(1)). - Daniel Hoyt, Apr 11 2020
From Stefano Spezia, Oct 26 2024: (Start)
2/(Pi - 2) = 1 + K_{n>=1} n*(n+1)/1, where K is the Gauss notation for an infinite continued fraction. In the expanded form, 2/(Pi - 2) = 1 + 1*2/(1 + 2*3/(1 + 3*4/(1 + 4*5/(1 + 5*6/(1 + ...))))) (see Finch at p. 23).
2/(Pi - 2) = Sum_{n>=1} (2/Pi)^n (see Shamos). (End)
Equals A309091/2. - Hugo Pfoertner, Oct 28 2024

cp's OEIS Frontend

A036245 Numerator of fraction equal to the continued fraction [ 0, 1, 4, ..., n^2 ].

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A036246 CONTINUANT transform of squares 1, 4, 9, ...

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

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A261827 Decimal expansion of the number whose continued fraction expansion consists of the perfect numbers (A000396).

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A329810 Decimal expansion of the constant whose continued fraction representation is [0; 1, 3, 7, 15, 31, ...] = A000225 (the Mersenne numbers).

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