A052119 -id:A052119 - cp's OEIS frontend

A385303 Decimal expansion of the real number whose continued fraction is Golomb's sequence (A001462).

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

Offset: 1

Jason Bard, Jun 24 2025

			1.4107845307495355919347994202105751786146865173661...

Jason Bard, Table of n, a(n) for n = 1..10000

Cf. A001462, A060997, A052119.

a[1] = 1; a[n_] := a[n] = 1 + a[n - a[a[n - 1]]]; (* A001462 *)
GenA385303[n_Integer] := Module[{cf1, cf2, d1, d2, i = n}, While[i < 2 n,
   cf1 = Table[a[k], {k, 1, i}]; cf2 = Table[a[k], {k, 1, i + 1}];
   d1 = RealDigits[FromContinuedFraction[cf1], 10, n+1][[1]]; d2 = RealDigits[FromContinuedFraction[cf2], 10, n+1][[1]];
   If[Take[d1,n] === Take[d2,n], Return[Take[d1,n]]]; i++;]];
GenA385303[100]

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

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A385303 Decimal expansion of the real number whose continued fraction is Golomb's sequence (A001462).

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

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