OFFSET
0,4
COMMENTS
Also the number of one-sided n-step prudent walks, starting from (0,0) and ending on the y-axis, with east, west and north steps. - Shanzhen Gao, Apr 26 2011
REFERENCES
T. S. Blyth, J. C. Varlet, Ockham algebras, Oxford Science Pub. 1994.
E. Munarini, Enumeration of order ideals of a garland, Ars Combin. 76 (2005), 185-192.
LINKS
Emanuele Munarini, Table of n, a(n) for n = 0..100
Aubrey Blecher and Arnold Knopfmacher, Prefixes of bargraph paths, Aequationes Math. (2023).
Shanzhen Gao and Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.
S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.
Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.
FORMULA
Recurrence: (n+6)*a(n+6) - (2*n+11)*a(n+5) - (n+3)*a(n+4) - 4*a(n+3) - (n+4)*c_(n+2) - (2*n+3)*a(n+1) + (n+1), a(n) = 0.
G.f.: (1 - x^2)/sqrt( 1 - 2*x - x^2 - x^4 + 2*x^5 + x^6 ).
a(n) = 1+sum(k=1..floor((n-1)/2), sum(i=1..min(n-2*k,k), C(n-2*k+1,i) * C(k-1,k-i) * C(n-k-i,k) ) ). - Shanzhen Gao, May 13 2011
EXAMPLE
a(4) = 7, since the central ideals of the garland G(4):
5..6..7..8
o..o..o..o
|\/|\/|\/|
|/\|/\|/\|
o..o..o..o
1..2..3..4
are: 1234, 1253, 1254, 1236, 2347, 1348, 2348.
a(4)=7, since there are 7 such walks: NNNN, NENW, NWNE, ENWN, ENNW, WNEN, WNNE. - Shanzhen Gao, May 13 2011
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Emanuele Munarini, Mar 21 2008
STATUS
approved