0,5

Number of Motzkin paths with no peaks and no valleys, i.e. no UD's and no DU's, where U=(1,1) and D=(1,-1). Example: a(7)=16 because there are 17 peakless Motzkin paths of length 6 (see A004148) of which only UHDUHD has a valley (here H=(1,0)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 08 2004

a(n+2) = number of Motzkin n-paths that avoid both UU and DD = number of Motzkin n-paths that avoid both UU and UFU. Example: a(7)=16 since of the 21 Motzkin 5-paths, only FUUDD, UFUDD, UUDDF, UUDFD, UUFDD contain a UU or DD (or both). Likewise, only FUUDD, UFUDD, UUDDF, UUDFD, UUFDD contain a UU or UFU. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004

The g.f. (z**3-z**2+2*z-1)*(z**3-z**2-2*z+1)/(2*z-1)/(2*z**5+z**3+z**2-3*z+1) conjectured by S. Plouffe in his 1992 dissertation is wrong.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. Doslic, D. Svrtan and D. Veljan, Enumerative aspects of secondary structures, Discr. Math., 285 (2004), 67-82.

Munarini, Emanuele, Combinatorial properties of the antichains of a garland. Integers, 9 (2009), 353-374.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

E. J. J. van Rensburg, Adsorbing bargraph paths in a q-wedge, Journal of Physics A, v.38 n.40, 8505-8525.

T. D. Noe, Table of n, a(n) for n=0..200

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

M. S. Waterman, Home Page (contains copies of his papers)

G.f.=2/[1-z+z^2+z^3+sqrt[(1-z^4)(1-2z-z^2)]]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 08 2004

G.f.: 1/(1-x-x^4/(1-x-x^2-x^3-x^4/(1-x-x^2-x^3-x^4/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), May 22 2009]

For Maple code see A004148.

a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-2-k ], {k, 2, n-2} ];

(PARI) a(n)=polcoeff((1-x+x^2+x^3-sqrt((1-x^4)*(1-2*x-x^2)+x^3*O(x^n)))/(2*x^2), n) /* Michael Somos Oct 28 2005 */

Third row of A064645.

Cf. A004148, A001006.

Sequence in context: A098588 A126683 A005821 this_sequence A129986 A110334 A084636

Adjacent sequences: A004146 A004147 A004148 this_sequence A004150 A004151 A004152

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).