1,2

This sequence arises in connection with mean lengths of ascents and descents in Dyck paths as follows. Let u(n,k) denote the mean length of the k-th ascent taken over all Dyck n-paths (A000108) where it is understood that if a Dyck path has fewer than k ascents, then the length of the k-th ascent is 0. For example, the second ascent in UUDUUUDDDDUD has length 3 and its fourth has length 0. Similarly, let v(n,k) denote the mean length of the k-th descent. Then u(k) := lim_{n->infty}u(n,k) and v(k) := lim_{n->infty}v(n,k) both exist. The sequence (u(k))_{k>=1} begins 3, 8/3, ... and decreases steadily toward a limit of 2. Analogously, v(k) increases steadily from 4/3 toward the same limit of 2. For all k>=1, u(k+1) exceeds 2 by the same amount that v(k) falls below 2. The common difference u(k+1) - 2 = 2 - v(k) is a(k+1)/3^(2k-1). Thus the common difference sequence begins 2/3, 14/27, 106/243,..., for k=1,2,3,... . - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006

Posting to newsgroup rec.puzzles, Dec 03 1999 by Nick Wedd (Nick(AT)maproom.co.uk).

Thread in newsgroup rec.puzzles, Dec 03 1999.

G.f.: {[x(1-x)] / [sqrt(1-10x+9x^2)] + x}/2. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 23 2004. Confirmed by Martin J. Erickson, Oct 05 2007.

a(1)=1; a(2)=2; a(n)=((10n - 16)a(n-1) - (9n - 27)a(n-2)) / (n-1), for n >= 3. - Martin J. Erickson (erickson(AT)truman.edu), Nov 12 2007

a(n) is asymptotic to (sqrt(2)/27)9^n/(sqrt(pi n)). - Martin J. Erickson, Nov 09 2007

Main diagonal of the square array given in A035002.

First differences of (A084771-1)/2.

Sequence in context: A122680 A121122 A026293 this_sequence A074618 A108436 A088754

Adjacent sequences: A051705 A051706 A051707 this_sequence A051709 A051710 A051711

easy,nonn,nice

Joe Keane (jgk(AT)jgk.org)

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 08 1999