cp's OEIS Frontend

Number of Motzkin paths of length n-1 with no peaks, no double rises and no doubledescents (i.e., no UD's, no UU's and no DD's, where U=(1,1) and D=(1,-1), n>0; can be easily formulated using RNA secondary structure terminology). E.g., a(5)=4 because we have HHHH, HUHD, UHDH and UHHD; here H=(1,0). Also number of peakless Motzkin paths of length n in which each D=(1,-1) step is followed by an H=(1,0) step (can be easily formulated using RNA secondary structure terminology). E.g., a(5)=4 because we have HHHHH, HUHDH, UHDHH and UHHDH (here U=(1,1)). - Emeric Deutsch, Jan 09 2004

The coefficient of t^n in the power series expansion of the solution u in the equation (1-t*u)(u-t*u-t-t^2*u^2+t^3*u)=0. - Shanzhen Gao, May 13 2011

Also the number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 3). The a(5) = 4 paths for n=5 are: UDUDUDUDUD, UUUUDDDDUD, UUUUDUDDDD, UDUUUUDDDD. - Alois P. Heinz, May 09 2012

a(n)=number of strictly alternating bargraphs of semiperimeter n+2. A bargraph is said to be strictly alternating if its ascents and descents alternate and all the formed peaks and valleys have width 1. An ascent (descent) is a maximal sequence of consecutive U (D) steps. Example: a(4) = 2 because among the 35 (=A082582(6)) bargraphs of semiperimeter 6 only those corresponding to the compositions [5] and [2,1,2] are strictly alternating. - Emeric Deutsch, Aug 26 2016

For n>=1, a(n) is the number of Dyck paths of semilength n+2 in which all ascent and descent lengths are >=3. For example, a(4) = 2 counts U^6.D^6, U^3.D^3.U^3.D^3 where ^ denotes repetition and a dot denotes concatenation. The gf F(x) = 1 + x^3 + x^4 + x^5 + 2*x^6 + ... for these paths satisfies F = 1 + x^3/(1-x) + (F-1)x^3/((1-x)(1-x*F)), which follows from a first return decomposition and summing over the lengths of the first ascent and first descent. A bijection to the title paths would be interesting. - David Callan, Dec 07 2021