A233588 -id:A233588 - cp's OEIS frontend

A233590 Decimal expansion of the continued fraction c(1) +c(1)/(c(2) +c(2)/(c(3) +c(3)/(c(4) +c(4)/....))), where c(i)=2^(i-1).

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

Offset: 1

Stanislav Sykora, Jan 06 2014

For more details about this type of continued fraction, see A233588.
This one corresponds to the powers of two sequence.
Corresponds to the regular continued fraction 1,2,2,4,4,8,8,16,16,... = A060546. - Jeffrey Shallit, Jun 14 2016

			1.408615979735005205132362590255795209484563373686888353703927022...

Stanislav Sykora, Table of n, a(n) for n = 1..20000 [a(1522) onward corrected by _Kevin Ryde_, Oct 26 2024]
Stanislav Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001
Stanislav Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki

Cf. A000079 (2^n), A096658, A060546.

Cf. Blazys's continued fractions: A233588, A233589, A233591 and Blazys' expansions: A233582, A233583, A233584, A233585, A233586, A233587

RealDigits[ Fold[(#2 + #2/#1) &, 1, Reverse@ (2^Range[0, 27])], 10, 111][[1]] (* Robert G. Wilson v, May 22 2014 *)

PARI
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Equals 1+1/(2+2/(4+4/(8+8/(16+16/(32+...))))).

Equals Product_{k>=0} ((1 - 2^(5*k + 2))*(1 - 2^(5*k + 3)))/((1 - 2^(5*k + 1))*(1 - 2^(5*k + 4))). - Antonio Graciá Llorente, Mar 20 2024

A233591 Decimal expansion of the continued fraction c(1) +c(1)/(c(2) +c(2)/(c(3) +c(3)/(c(4) +c(4)/....))), where c(i)=i^2.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jan 06 2014

For more details on this type of continued fractions, see A233588.

This one corresponds to the squares of natural numbers.

			1.22628402418269027481493710086224039619081148735362359550166652...

Stanislav Sykora, Table of n, a(n) for n = 1..20000 [a(322) onward corrected by _Kevin Ryde_, Oct 28 2024]
Stanislav Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001
Stanislav Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki

Cf. A000290 (n^2).

Cf. Blazys' continued fractions: A233588, A233589, A233591 and Blazys' expansions: A233582, A233583, A233584, A233585, A233586, A233587.

 RealDigits[ Fold[(#2 + #2/#1) &, 1, Reverse@ Range[45]^2], 10, 111][[1]] (* Robert G. Wilson v, May 22 2014 *)

PARI
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See the link
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Equals 1+1/(4+4/(9+9/(16+16/(25+25/(36+...))))).

A330867 Decimal expansion of the continued fraction 1/(1 + 2/(2 + 3/(3 + 5/(5 + 7/(7 + ... + prime(k)/(prime(k) + ...)))))).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 28 2020

			0.58152500459221465439915170481800446195586754...

Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A000040, A064442, A068996, A073333, A084255, A085825, A152062, A191504, A191815, A233588, A247856.

cp's OEIS Frontend

A233590 Decimal expansion of the continued fraction c(1) +c(1)/(c(2) +c(2)/(c(3) +c(3)/(c(4) +c(4)/....))), where c(i)=2^(i-1).

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A233591 Decimal expansion of the continued fraction c(1) +c(1)/(c(2) +c(2)/(c(3) +c(3)/(c(4) +c(4)/....))), where c(i)=i^2.

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A330867 Decimal expansion of the continued fraction 1/(1 + 2/(2 + 3/(3 + 5/(5 + 7/(7 + ... + prime(k)/(prime(k) + ...)))))).

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