0,2

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 (deutsch(AT)duke.poly.edu), Aug 29 2004

E.g.f.=[3-x-2(1+x)exp(-x)]/(1-x)^3. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 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.

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);

Cf. A000255, A096655, A096656, A096657.

Adjacent sequences: A096651 A096652 A096653 this_sequence A096655 A096656 A096657

Sequence in context: A020031 A001340 A058786 this_sequence A060389 A101714 A077318

nonn

Clark Kimberling (ck6(AT)evansville.edu), Jul 01 2004

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 29 2004