A233583 -id:A233583 - 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 *)

PARI
```
See the link.
```

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

A233582 Coefficients of the generalized continued fraction expansion Pi = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

Original entry on oeis.org

3, 21, 111, 113, 158, 160, 211, 216, 525, 1634, 1721, 7063, 8771, 15077, 26168, 58447, 223767, 254729, 587278, 1046086, 1491449, 1635223, 1689171, 2039096, 2290214, 13444599, 22666443, 1276179737, 4470200748

Offset: 1

Stanislav Sykora, Jan 02 2014

Definition of "Blazys" generalized continued fraction expansion of an irrational real number x>1:
Set n=1,r=x; (ii) set a(n)=floor(r); (iii) set r=a(n)/(r-a(n)); (iv) increment n and iterate from point (ii).
For the inverse of this mapping, see A233588.

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. A000796 (Pi), A233583-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[Pi, 33] (* Robert G. Wilson v, May 22 2014 *)

bx(x,nmax)={local(c,v,k);
v = vector(nmax);c = x;for(k=1,nmax,v[k] = floor(c);c = v[k]/(c-v[k]););return (v);}
bx(Pi,1000) \\ Execution; use very high real precision

Pi = 3+3/(21+21/(111+111/(113+113/(158+...)))).

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+...))))))).

A233587 Coefficients of the generalized continued fraction expansion sqrt(7) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

Original entry on oeis.org

2, 3, 30, 34, 111, 235, 3775, 5052, 7352, 9091, 34991, 35530, 53424, 57290, 66023, 1409179, 1519111, 1725990, 1812396, 4370835, 4507156, 4655396, 44257080, 234755198, 261519946, 264374278, 273487975

Offset: 1

Stanislav Sykora, Jan 06 2014

nonn

For more details on Blazys' expansions, see A233582.

Sqrt(7) is the first square root of a natural number with an a-periodic Blazys' expansion (see A233592 and A233593).

Stanislav Sykora, Table of n, a(n) for n = 1..1000

Cf. Blazys' expansions: A233582 (Pi), A233583 (e), A233584 (sqrt(e)), A233585 (1/gamma), A233585 (2*gamma) and Blazys' continued fractions: A233588, A233589, A233590, A233591.

Cf. A010465, A010121.

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@7, 32] (* 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(sqrt(7), 1000) \\ Execution; use very high real precision

sqrt(7) = 2+2/(3+3/(30+30/(34+34/(111+...)))).

A233586 Coefficients of the generalized continued fraction expansion of twice the Euler constant, 2*gamma = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

Original entry on oeis.org

1, 6, 12, 19, 63, 263, 856, 2632, 7714, 9683, 888970, 1200867, 1691244, 2350415, 3433770, 4482812, 17544235, 48509602, 53801529, 114221223, 124712727, 997393454, 16681741997, 17954856574, 105651203040

Offset: 1

Stanislav Sykora, Jan 06 2014

nonn

For more details on Blazys' expansions, see A233582.

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. A001620 (gamma).

Cf. Blazys' expansions: A233582 (Pi), A233583 (e), A233584 (sqrt(e)), A233585 (1/gamma), 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[2 EulerGamma, 29] (* 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(2*Euler, 670) \\ Execution; use very high real precision

2*gamma = 1+1/(6+6/(12+12/(19+19/(63+63/(263+...))))).

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
```
See the link.
```

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
```
See the link
```

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
```
See the link
```

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

A254667 The nonnegative numbers with 2 instead of 1.

Original entry on oeis.org

0, 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, 57, 58, 59, 60, 61, 62

Offset: 0

Paul Curtz, Feb 04 2015

An autosequence of the first kind is a sequence whose main diagonal in the difference table is A000004 = 0's.
This is the case for a(n).
Difference table of a(n):
   0,  2,  2,  3,  4,  5, ...
   2,  0,  1,  1,  1,  1, ...
  -2,  1,  0,  0,  0,  0, ...
   3, -1,  0,  0,  0,  0, ...
  -4,  1,  0,  0,  0,  0, ...
   5, -1,  0,  0,  0,  0, ...
  etc.
The inverse binomial transform of a(n) is (-1)^(n+1)*a(n).
0 followed by A000012(n) is not in the OEIS. See A054977.
What is the meaning of a(n)?
Among many others, A015441 is an autosequence of the first kind.
General form for such autosequence.
Starting from the first upper diagonal s0, s1, s2, s3, s4, ...,
the autosequence is
0, s0, s0, s0 + s1, s0 + 2*s1, s0 + 3*s1 + s2, s0 + 4*s1 + 3*s2, ... .
After 0, the corresponding coefficients are A011973(n).

			G.f. = 2*x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 8*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000004, A001477, A011973, A015441, A027641, A027642, A054977, A063524, A164555, A164558, A233583.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2*x-2*x^2+x^3)/(1-x)^2)); // G. C. Greubel, Aug 03 2018

CoefficientList[Series[(2*x-2*x^2+x^3)/(1-x)^2, {x, 0, 60}], x] (* G. C. Greubel, Aug 03 2018 *)
a[ n_] := n + Boole[n == 1]; (* Michael Somos, Aug 19 2018 *)
Join[{0,2},Range[2,70]] (* Harvey P. Dale, Oct 10 2024 *)

{a(n) = n + (n==1)}; /* Michael Somos, Feb 09 2015 */

a(n) = (A164558(n) + (-1)^(n+1)*A164555(n))/A027642(n).
a(n) = A063524(n) + A001477(n). - David A. Corneth, Aug 03 2018
G.f.: (2*x - 2*x^2 + x^3) / (1 - x)^2. - Michael Somos, Feb 09 2015

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)/...))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A233582 Coefficients of the generalized continued fraction expansion Pi = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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)/....))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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)/....))).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A233587 Coefficients of the generalized continued fraction expansion sqrt(7) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A233586 Coefficients of the generalized continued fraction expansion of twice the Euler constant, 2*gamma = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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).

Views