[seqfan] Two neighbors sum -- and odd ranks in S

[Eric Angelini]

Hello SeqFans,
S is the lexically first permutation 
of the Natural numbers such that 
a(n)+a(n+1)=a(2n-1)

S = 1,0,2,3,5,4,8,6,9,7,12,10,14,11,15,13,16,17,19,18,22,20,24,21,25,23,26,27,28,30,29,31,33,32,36,34,37,35,40,38,42,39,44,41,45,43,46,47,48,50,49,51,53,52,55,54,...

Take n=1, then:
a(n)=a(1)=1
a(n+1)=a(2)=0
a(2n-1)=a(1)=1 and 1+0 is indeed 1

Take n=6, then:
a(n)=a(6)=4
a(n+1)=a(7)=8
a(2n-1)=a(11)=12 and 4+8 is indeed 12

Take n=21, then:
a(n)=a(21)=22
a(n+1)=a(22)=20
a(2n-1)=a(41)=42 and 22+20 is indeed 42

Best,
É.