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A256210 Lexicographically earliest permutation of positive integers starting with 2 such that a(a(n)+a(n+1)) is odd for all n.

%I A256210 #21 Aug 04 2018 10:47:12
%S A256210 2,1,3,5,4,6,7,9,11,13,8,10,15,12,14,17,16,19,18,21,23,20,22,25,27,29,
%T A256210 31,24,26,28,33,30,35,32,37,34,39,36,41,38,40,43,45,47,42,44,49,46,48,
%U A256210 51,50,53,52,55,57,59,54,56,58,61,63,60,65,62,67,64,69
%N A256210 Lexicographically earliest permutation of positive integers starting with 2 such that a(a(n)+a(n+1)) is odd for all n.
%C A256210 This is the sequence U defined in the comments on A255003.
%H A256210 Alois P. Heinz, <a href="/A256210/b256210.txt">Table of n, a(n) for n = 1..20000</a>
%H A256210 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%p A256210 N:= 100: # to get a(n) for n <= N
%p A256210 maxodd:= -1:
%p A256210 maxeven:= 2:
%p A256210 a[1]:= 2:
%p A256210 needodd:= {}:
%p A256210 for n from 2 to N do
%p A256210   if member(n,needodd) or maxodd < maxeven then
%p A256210      a[n]:= maxodd + 2;
%p A256210      maxodd:= a[n];
%p A256210   else
%p A256210      a[n]:= maxeven + 2;
%p A256210      maxeven:= a[n];
%p A256210   fi;
%p A256210   needodd:= needodd union {a[n-1]+a[n]};
%p A256210 od:
%p A256210 seq(a[n],n=1..N); # _Robert Israel_, Mar 26 2015
%t A256210 nn = 100;
%t A256210 maxodd = -1;
%t A256210 maxeven = 2;
%t A256210 a[1] = 2;
%t A256210 needodd = {};
%t A256210 For[n = 2, n <= nn, n++,
%t A256210     If[MemberQ[needodd, n] || maxodd < maxeven,
%t A256210        a[n] = maxodd + 2;
%t A256210        maxodd = a[n]
%t A256210    ,
%t A256210        a[n] = maxeven + 2;
%t A256210        maxeven = a[n]
%t A256210    ];
%t A256210     needodd = needodd ~Union~ {a[n-1]+a[n]};
%t A256210 ];
%t A256210 Array[a, nn] (* _Jean-François Alcover_, Aug 04 2018, after _Robert Israel_ *)
%o A256210 (Haskell)  after Robert Israel's Maple program
%o A256210 import Data.IntSet (empty, member, insert)
%o A256210 a256210 n = a256210_list !! (n-1)
%o A256210 a256210_list = 2 : f [2 ..] 2 [1, 3 ..] [4, 6 ..] empty where
%o A256210    f (x:xs) y us'@(u:us) vs'@(v:vs) s
%o A256210      | member x s || u < v = u : f xs u us vs' (insert (y + u) s)
%o A256210      | otherwise           = v : f xs v us' vs (insert (y + v) s)
%o A256210 -- _Reinhard Zumkeller_, Mar 26 2015
%Y A256210 Cf. A255003.
%Y A256210 Cf. A256371 (inverse), A256372 (fixed points).
%K A256210 nonn
%O A256210 1,1
%A A256210 _N. J. A. Sloane_, Mar 26 2015