A233590 -id:A233590 - 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
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See the link.
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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+...))))))).

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
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See the link.
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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

A096654 Denominators of self-convergents to 1/(e-2).

Original entry on oeis.org

1, 2, 8, 38, 222, 1522, 11986, 106542, 1054766, 11506538, 137119578, 1772006854, 24681524038, 368577425634, 5874202721042, 99515904921182, 1785757627196766, 33835407673201882, 675016383080377546, 14143200407398386678, 310507536216973671158, 7128173005328786885714

Offset: 0

Clark Kimberling, Jul 01 2004

The self-continued fraction of r>0 is here introduced as the sequence (b(0), b(1), b(2), ...) defined as follows: put r(0)=r, b(0)=[r(0)] and for n>=1, put r(n)=b(n-1)/(r(n-1)-b(n-1)) and b(n)=[r(n)]. This differs from simple continued fraction, for which r(n)=1/(r(n-1)-b(n-1)). Now r=lim(p(n)/q(n)), where p(0)=b(1), q(0)=1, p(1)=b(0)(b(1)+1), q(1)=b(1) and for n>=2, p(n)=b(n)*p(n-1)+b(n-1)*p(n-2), q(n)=b(n)*q(n-1)+b(n-1)*q(n-2); p(0),p(1),... are the numerators of the self-convergents to r; q(0),q(1),... are the denominators of the self-convergents to r. Thus A096654 is given by a(n)=(n+1)*a(n-1)+n*a(n-2), a(0)=1, a(1)=2.

Number of increasing runs of odd length in all permutations of [n+1]. Example: a(2) = 8 because we have (123), 13(2), (3)12, (2)13, 23(1), (3)(2)(1) (the runs of odd length are shown between parentheses). - Emeric Deutsch, Aug 29 2004

			a(2)=q(2)=3*2+2*1=8, a(3)=q(3)=4*8+3*2=38. The convergents p(0)/q(0) to p(4)/q(4) are 1/1, 3/2, 11/8, 53/38, 309/222.

Cf. A000255, A096655, A096656, A096657, A096658, A097591, A233590.

G:=(3-x-2*(1+x)*exp(-x))/(1-x)^3: Gser:=series(G,x=0,22): 1,seq(n!*coeff(Gser,x^n),n=1..21);

With[{g = (3 - x - 2*(1 + x)*Exp[-x])/(1 - x)^3},CoefficientList[Series[g, {x, 0, 21}], x]*Table[k!, {k, 0, 21}]] (* Shenghui Yang, Oct 15 2024 *)

x='x+O('x^66); Vec(serlaplace((3-x-2*(1+x)*exp(-x))/(1-x)^3)) /* Joerg Arndt, Aug 06 2012 */

prpr = 1
prev = 2
for n in range(2, 77):
    print(prpr, end=', ')
    curr = (n+1)*prev + n*prpr
    prpr = prev
    prev = curr
# Alex Ratushnyak, Aug 05 2012

a(n) = (n+1)*a(n-1) + n*a(n-2), with a(0)=1, a(1)=2. - Alex Ratushnyak, Aug 05 2012
E.g.f.: (3-x-2*(1+x)*exp(-x))/(1-x)^3. - Emeric Deutsch, Aug 29 2004
From Gary Detlefs, Apr 12 2010: (Start)
a(n) = A055596(n+1) + A055596(n+2).
a(n) = (n+1)!+(n+2)! -2*( A000166(n+1) + A000166(n+2)).
a(n) = (n+1)! - 2*floor(((n+1)!+1)/e) + (n+2)!-2*floor(((n+2)!+1)/e). (End)
a(n) = ((n+3)!-2*floor(((n+3)!+1)/e))/(n+2). - Gary Detlefs, Jul 11 2010 [corrected by Gary Detlefs, Oct 26 2020]
a(n) = Sum_{k=1..n+1} A097591(n+1,k). - Alois P. Heinz, Jul 03 2019

More terms from Emeric Deutsch, Aug 29 2004

A096658 a(n) = (2^n)a(n-1) + (2^(n-1))a(n-2), a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 10, 88, 1488, 49024, 3185152, 410836992, 105581969408, 54163142606848, 55517115997749248, 113754516621419872256, 466052199134899187220480, 3818365553813175477506932736, 62563919133290380117615296118784

Offset: 0

Clark Kimberling, Jul 01 2004

nonn

This is the sequence of denominators of self-convergents to the number 1.40861... (see A233590) whose self-continued fraction is (1,2,4,8,16,...). See A096657 for numerators and A096654 for definitions.

Harvey P. Dale, Table of n, a(n) for n = 0..81

Cf. A000079, A096654, A096657, A233590.

a[0]=1; a[1]=2; a[n_] := (2^n)*a[n-1] + (2^(n-1))*a[n-2]; Table[ a[n], {n, 0, 14}] (* Robert G. Wilson v, Jul 03 2004 *)
RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2^n a[n-1]+2^(n-1) a[n-2]},a,{n,20}] (* Harvey P. Dale, Feb 16 2020 *)

a(n) is asymptotic to c*2^(n(n+1)/2) where c=1.54241381761010214381886547... - Benoit Cloitre, Jul 01 2004

c = (1 + Sum_{k>=1} (Product_{j=1..k} 1/(2^(j-1)*(2^j-1)))) / A233590 = 1.5424138176101021438188654719396629292944606799275904286064... . - Vaclav Kotesovec, Nov 27 2015

More terms from Benoit Cloitre, Jul 02 2004

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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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)/....))).

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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)/....))).

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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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A096654 Denominators of self-convergents to 1/(e-2).

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A096658 a(n) = (2^n)a(n-1) + (2^(n-1))a(n-2), a(0)=1, a(1)=2.