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A096654 Denominators of self-convergents to 1/(e-2). 6
1, 2, 8, 38, 222, 1522, 11986, 106542, 1054766, 11506538, 137119578, 1772006854, 24681524038, 368577425634, 5874202721042, 99515904921182, 1785757627196766, 33835407673201882, 675016383080377546, 14143200407398386678 (list; graph; refs; listen; history; text; internal format)
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..19.

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

Contribution 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). - Gary Detlefs, Jul 11 2010

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,

    curr = (n+1)*prev + n*prpr

    prpr = prev

    prev = curr

# from 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, A233590.

Sequence in context: A001340 A275707 A058786 * A269509 A191016 A265906

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

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Last modified June 23 21:29 EDT 2017. Contains 288675 sequences.