A233586 -id:A233586 - cp's OEIS frontend

A233588 Decimal expansion of the continued fraction prime(1) + prime(1)/(prime(2) + prime(2)/(prime(3) + prime(3)/(prime(4) + prime(4)/...))).

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

Offset: 1

Stanislav Sykora, Jan 02 2014

Given a nondecreasing sequence s(n) of natural numbers (such as, in this case, that of primes A000079), the corresponding continued fraction is bf(s(n)) = a(1) + a(1)/(a(2) + a(2)/(a(3) + a(3)/(...))).

For the inverse of this mapping of nondecreasing sequences of natural numbers into irrational real numbers greater than 1, see A233582.

			2.56654383217138884446752910633228575178297282870231464596973352546639971...

Stanislav Sykora, Table of n, a(n) for n = 1..25000 [a(224) onward corrected by _Kevin Ryde_, Oct 27 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 (primes), A233589, A233590, A233591, A233582, A233583, A233584, A233585, A233586, A233587.

RealDigits[ Fold[(#2 + #2/#1) &, 1, Reverse@ Prime@ Range@ 57], 10, 111][[1]] (* Robert G. Wilson v, May 22 2014 *)

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Equals 2 + 2/(3 + 3/(5 + 5/(7 + 7/(11 + 11/(13 + 13/(17 + ...)))))).

A233583 Coefficients of the generalized continued fraction expansion e = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

Original entry on oeis.org

2, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56

Offset: 1

Stanislav Sykora, Jan 06 2014

For more details on Blazys' expansion, see A233582.

This sequence matches that of natural numbers (A000027), offset by 1, with two different starting terms.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
S. Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001
S. Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki

Cf. A000027 (natural numbers), A001113 (number e).

Cf. Blazys' expansions: A233582 (Pi), A233584, A233585, A233586, A233587 and Blazys' continued fractions: A233588, A233589, A233590, A233591.

BlazysExpansion[n_, mx_] := Block[{k = 1, x = n, lmt = mx + 1, s, lst = {}}, While[k < lmt, s = Floor[x]; x = 1/(x/s - 1); AppendTo[lst, s]; k++]; lst]; BlazysExpansion[E, 80] (* Robert G. Wilson v, May 22 2014 *)

default(realprecision,100);
bx(x,nmax)={local(c,v,k); \\ Blazys expansion function
v = vector(nmax);c = x;for(k=1,nmax,v[k] = floor(c);c = v[k]/(c-v[k]););return (v);}
bx(exp(1),100) \\ Execution; use high real precision

e = 2+2/(2+2/(2+2/(3+3/(4+4/(5+...))))).

A233584 Coefficients of the generalized continued fraction expansion sqrt(e) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

Original entry on oeis.org

1, 1, 1, 1, 5, 9, 17, 109, 260, 2909, 3072, 3310, 3678, 6715, 35175, 37269, 439792, 1400459, 1472451, 4643918, 5683171, 44850176, 62252861, 145631385, 154435765, 371056666, 1685980637, 11196453405, 14795372939

Offset: 1

Stanislav Sykora, Jan 06 2014

nonn

For more details on Blazys' expansions, see A233582.

Compared with simple continued fraction expansion for sqrt(e), this sequence starts soon growing very rapidly.

Stanislav Sykora, Table of n, a(n) for n = 1..1000
S. Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001
S. Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki

Cf. A019774 (sqrt(e)), A058281 (simple continued fraction).

Cf. Blazys' expansions: A233582 (Pi), A233583, A233585, A233586, A233587 and Blazys' continued fractions: A233588, A233589, A233590, A233591.

BlazysExpansion[n_, mx_] := Block[{k = 1, x = n, lmt = mx + 1, s, lst = {}}, While[k < lmt, s = Floor[x]; x = 1/(x/s - 1); AppendTo[lst, s]; k++]; lst]; BlazysExpansion[Sqrt@E, 35] (* Robert G. Wilson v, May 22 2014 *)

bx(x, nmax)={local(c, v, k); \\ Blazys expansion function
v = vector(nmax); c = x; for(k=1, nmax, v[k] = floor(c); c = v[k]/(c-v[k]); ); return (v); }
bx(exp(1/2), 100) \\ Execution; use high real precision

sqrt(e) = 1+1/(1+1/(1+1/(1+1/(5+5/(9+9/(17+17/(109+...))))))).

A233585 Coefficients of the generalized continued fraction expansion of the inverse of Euler constant, 1/gamma = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 4, 12, 39, 71, 83, 484, 1028, 1447, 9913, 31542, 526880, 685669, 1396494, 1534902, 2295194, 9521643, 9643315, 42421746, 183962859, 553915624, 557976754, 6111180351, 10671513549, 61650520975, 106532505646

Offset: 1

Stanislav Sykora, Jan 06 2014

nonn

Stanislav Sykora, Table of n, a(n) for n = 1..670
S. Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001
S. Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki

Cf. A233582.
Cf. A001620 (gamma).
Cf. Blazys's expansions: A233582 (Pi), A233583(e), A233584 (sqrt(e)), A233586 (2*gamma), A233587 and Blazys's continued fractions: A233588, A233589, A233590, A233591.

BlazysExpansion[n_, mx_] := Block[{k = 1, x = n, lmt = mx + 1, s, lst = {}}, While[k < lmt, s = Floor[x]; x = 1/(x/s - 1); AppendTo[lst, s]; k++]; lst]; BlazysExpansion[1/EulerGamma, 35] (* Robert G. Wilson v, May 22 2014 *)
BlazysExpansion[n_, mx_] := Reap[Nest[(1/(#/Sow[Floor[#]] - 1)) &, n, mx];][[-1, 1]]; BlazysExpansion[1/EulerGamma, 35] (* Jan Mangaldan, Jan 04 2017 *)

bx(x, nmax)={local(c, v, k); \\ Blazys expansion function
v = vector(nmax); c = x; for(k=1, nmax, v[k] = floor(c); c = v[k]/(c-v[k]); ); return (v); }
bx(1/Euler, 670) \\ Execution; use very high real precision

1/gamma = 1+1/(1+1/(2+2/(2+2/(2+2/(2+2/(4+4/(12+...))))))).

A233589 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-1)!.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jan 06 2014

			1.69880476767000721195269011591464043255973093664983969781741917426892...

Stanislav Sykora, Table of n, a(n) for n = 1..20000 (terms 6942..20000 corrected by Amiram Eldar)
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. A233588.
Cf. A000142 (factorials), A006882 (double factorials).
Cf. Blazys' continued fractions: A233588, A233590, A233591 and Blazys' expansions: A233582, A233583, A233584, A233585, A233586, A233587.

RealDigits[ Fold[(#2 + #2/#1) &, 1, Reverse@ Range[0, 18]!], 10, 111][[1]] (* Robert G. Wilson v, May 22 2014 *)

PARI
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Equals 0!+0!/(1!+1!/(2!+2!/(3!+3!/(4!+...)))).

Equals simple continued fraction [0!!; 1!!, 2!!, 3!!, ..., n!!, ...] where the double factorial n!! = A006882(n). - Thomas Ordowski, Oct 21 2024

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).

Original entry on oeis.org

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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Equals 1+1/(4+4/(9+9/(16+16/(25+25/(36+...))))).

A233592 Positive integers k such that the continued fraction expansion sqrt(k) = c(1) + c(1)/(c(2) + c(2)/(c(3) + c(3)/...)) is periodic.

Original entry on oeis.org

2, 3, 5, 6, 8, 10, 11, 12, 15, 17, 18, 20, 24, 26, 27, 30, 35, 37, 38, 39, 40, 42, 44, 45, 48, 50, 51, 56, 63, 65, 66, 68, 72, 80, 82, 83, 84, 87, 90, 99, 101, 102, 104, 105, 108, 110, 120, 122, 123, 132, 143, 145

Offset: 1

Stanislav Sykora, Jan 06 2014

nonn

For more details on this type of expansion, see A233582.
The cases with aperiodic expansions are listed in A233593.
All the listed cases become periodic after just two leading terms (it is a conjecture that this behavior is general); the validity of their expansions was explicitly tested.

			Blazys's expansion of sqrt(2) is {1, 2, 4, 4, 4, 4, 4, ...}, i.e., it has a periodic termination. Consequently, 2 is a term of this sequence.

Stanislav Sykora, Table of n, a(n) for n = 1..200
Stanislav Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013.
Stanislav Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki.

Cf. A233593.

Cf. Blazys's expansions: A233582, A233584, A233585, A233586, A233587.

PARI
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A233593 Positive integers k such that the continued fraction expansion sqrt(k) = c(1) + c(1)/(c(2) + c(2)/(c(3) + c(3)/....)) is aperiodic.

Original entry on oeis.org

7, 13, 14, 19, 21, 22, 23, 28, 29, 31, 32, 33, 34, 41, 43, 46, 47, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 85, 86, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 103, 106, 107

Offset: 1

Stanislav Sykora, Jan 06 2014

nonn

For more details about this type of expansions, see A233582.

The cases with known periodic expansions, listed in A233592, all become periodic after just two leading terms. In contrast, the Blazys's expansion of sqrt(a(k)) for every member a(k) of this list remains aperiodic up to at least 1000 terms. It is therefore conjectured, though not proved, that these expansions are indeed aperiodic.

			Blazys's expansion of sqrt(7), A233587, is {2, 3, 30, 34, 111, ...}. Its first 1000 terms are all distinct. Hence, 7 is a term of this sequence.

Stanislav Sykora, Table of n, a(n) for n = 1..200
Stanislav Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013.
Stanislav Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki.

Cf. A233592.

Cf. Blazys's expansions: A233582, A233584, A233585, A233586, A233587.

cp's OEIS Frontend

A233588 Decimal expansion of the continued fraction prime(1) + prime(1)/(prime(2) + prime(2)/(prime(3) + prime(3)/(prime(4) + prime(4)/...))).

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A233583 Coefficients of the generalized continued fraction expansion e = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

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A233584 Coefficients of the generalized continued fraction expansion sqrt(e) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

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A233585 Coefficients of the generalized continued fraction expansion of the inverse of Euler constant, 1/gamma = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

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A233589 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-1)!.

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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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