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

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

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..20.

Formula

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

Example

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.

Maple

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);

Prog

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

Crossrefs

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

Sequence in context: A001340 A275707 A058786 * A371312 A269509 A307725

Adjacent sequences: A096651 A096652 A096653 * A096655 A096656 A096657

Keyword

nonn,frac

Author

Clark Kimberling, Jul 01 2004

Extensions

More terms from Emeric Deutsch, Aug 29 2004

Status

approved