OFFSET
0,4
COMMENTS
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.
Also permutations of {1...n} matching the consecutive patterns (1,2,3) or (3,2,1). Matching only one of these gives A065429.
LINKS
FORMULA
a(n) = n! - A001250(n).
EXAMPLE
The a(4) = 14 permutations:
(1,2,3,4) (3,1,2,4)
(1,2,4,3) (3,2,1,4)
(1,3,4,2) (3,4,2,1)
(1,4,3,2) (4,1,2,3)
(2,1,3,4) (4,2,1,3)
(2,3,4,1) (4,3,1,2)
(2,4,3,1) (4,3,2,1)
MAPLE
b:= proc(u, o) option remember;
`if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
end:
a:= n-> n!-`if`(n<2, 1, 2)*b(n, 0):
seq(a(n), n=0..30); # Alois P. Heinz, Nov 04 2021
MATHEMATICA
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]] ==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Permutations[Range[n]], !wigQ[#]&]], {n, 0, 6}]
PROG
(Python)
from itertools import accumulate, count, islice
def A348615_gen(): # generator of terms
yield from (0, 0)
blist, f = (0, 2), 1
for n in count(2):
f *= n
yield f - (blist := tuple(accumulate(reversed(blist), initial=0)))[-1]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 03 2021
STATUS
approved