OFFSET
0,3
COMMENTS
Suppose a < b, c < b, and c < d. To avoid abcd means not to have four consecutive letters such that the first letter is less than the second one, the third letter is less than the second one, and the third letter is less than the last one.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..468
Sergey Kitaev, Introduction to partially ordered patterns, Discrete Applied Mathematics 155 (2007), 929-944.
FORMULA
a(n) ~ c * d^n * n!, where d = A245758 = 0.7827041801715217018447074977..., c = 2.035127405829990832658061124449458067... . - Vaclav Kotesovec, Aug 22 2014
MAPLE
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(b(u-j, o+j-1, [1, 3, 1][t]), j=1..u)+
`if`(t=3, 0, add(b(u+j-1, o-j, 2), j=1..o)))
end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25); # Alois P. Heinz, Mar 10 2020
MATHEMATICA
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
Sum[b[u - j, o + j - 1, {1, 3, 1}[[t]]], {j, 1, u}] +
If[t == 3, 0, Sum[b[u + j - 1, o - j, 2], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 08 2022, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Submitted independently by Signy Olafsdottir (signy06(AT)ru.is), May 09 2010 (9 terms) and R. H. Hardin, May 10 2010 (17 terms)
EXTENSIONS
a(18)-a(23) from Alois P. Heinz, Oct 06 2013
a(0)=1 prepended by Alois P. Heinz, Mar 10 2020
STATUS
approved