A096658 -id:A096658 - 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

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

A096657 a(n) = (2^n)a(n-1) + (2^(n-1))a(n-2), a(0)=1, a(1)=3.

Original entry on oeis.org

1, 3, 14, 124, 2096, 69056, 4486656, 578711552, 148724449280, 76295068188672, 78202296743231488, 160236429879963287552, 656488575092059763900416, 5378610735570941915498020864, 88128536246001466497105446043648

Offset: 0

Clark Kimberling, Jul 01 2004

nonn

This is the sequence of numerators of self-convergents to the number 1.40861... whose self-continued fraction is (1,2,4,8,16,...)=A000079. See A096658 for denominators and A096654 for definitions.

			a(2)=4*3+2*1=14, a(3)=8*14+4*3=124.

Cf. A000079, A096654, A096658.

a[0] = 1; a[1] = 3; 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 *)
b[n_, k_] := k^2 - k (1 + n) +  n (1 + n)/2;
a[n_] := Sum[2^b[n, k] QBinomial[n - k + 1, k, 2], {k, 0, n + 1}] ;
Table[a[n], {n, 0, 14}] (* After Vladimir Kruchinin, Peter Luschny, Jan 19 2020 *)

a(n) is asymptotic to c*2^(n(n+1)/2) where c = 2.1726687508496636560169136... - Benoit Cloitre, Jul 02 2004
c = 1 + Sum_{k>=1} (Product_{j=1..k} 1/(2^(j-1)*(2^j-1))) = 2.172668750849663656016913609859312820656436935109608860295... . - Vaclav Kotesovec, Nov 27 2015
a(n) = Sum_{k=0..n+1} q-binomial(n-k+1,k)*2^(binomial(n-k+1,2)+binomial(k,2)), where q-binomial is triangle A022166, that is, with q=2. - Vladimir Kruchinin, Jan 19 2020

More terms from Benoit Cloitre, Jul 02 2004

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

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A096657 a(n) = (2^n)a(n-1) + (2^(n-1))a(n-2), a(0)=1, a(1)=3.

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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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Mathematica

PARI

Formula

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

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A096657 a(n) = (2^n)*a(n-1) + (2^(n-1))*a(n-2), a(0)=1, a(1)=3.

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Extensions

A096657 a(n) = (2^n)a(n-1) + (2^(n-1))a(n-2), a(0)=1, a(1)=3.