# [seqfan] Re: Permutations with even sums of adjacent triples

Rob Pratt Rob.Pratt at sas.com
Mon Dec 18 00:52:35 CET 2017

```I confirm the values you reported, and I guess you could include a[1] = 1 and a[2] = 2 (vacuously).

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of hv at crypt.org
Sent: Sunday, December 17, 2017 4:44 PM
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Cc: seqfan at seqfan.eu
Subject: [seqfan] Re: Permutations with even sums of adjacent triples

EXTERNAL

"Richard J. Mathar" <mathar at mpia-hd.mpg.de> wrote:
: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)

I think it will be 0 for all n > 7.

In any permutation P = [ p_1 .. p_n ] we need p_i == p_{i+3} (mod 2) for 1 <= i <= n - 3. We also need each triple to have one even and two odd elements.

The last opportunity for such a pattern is 1 .. 7 => EOOEOOE; after that we need too many odd numbers.

Hugo

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