[seqfan] Permutations with even sums of adjacent triples

[Richard J. Mathar]

Les Reid asks in http://people.missouristate.edu/lesreid/POW12_1011.html:
how many permutations of [n] exist such that the sum of 3 adjacent
numbers is always even? This seems to start

6,4,24,0,144,0,0,0 (n>=3)

computed with a naive Maple program:

is3ev := proc(L)
        for s from 1 to nops(L)-2 do
                if type(op(s,L)+op(s+1,L)+op(s+2,L),'odd') then
                        return false;
                end if;
        end do:
        true ;
end proc:

P3ev := proc(n)
        a := 0 ;
        if n >= 3 then
                for p in combinat[permute](n) do
                        if is3ev(p) then
                                a := a+1 ;
                        end if;
                end do:
        end if:
        a ;
end proc:


Example: a(n=3)=6 because for all 6 permutations the sums are 6.
a(n=4)=4 counts [2, 1, 3, 4], [2, 3, 1, 4], [4, 1, 3, 2], and [4, 3, 1, 2]. 
Someone with a good feeling of recursive enumeration might extend
and submit this.....

RJM