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a(1) = 0, and for any n > 0, a(2*n) = a(n) + k(n) and a(2*n+1) = a(n) + 3 * k(n) where k(n) is the least positive integer not leading to a duplicate term in the sequence.
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%I #46 Dec 21 2018 14:55:10

%S 1,2,4,3,5,6,10,7,15,8,14,11,21,12,16,9,13,18,24,17,35,19,29,20,38,23,

%T 27,22,42,25,43,26,60,28,58,30,54,31,45,32,62,37,41,33,61,34,44,36,68,

%U 47,65,39,71,40,66,46,94,49,63,50,100,51,67,48,92,64,72

%N a(1) = 0, and for any n > 0, a(2*n) = a(n) + k(n) and a(2*n+1) = a(n) + 3 * k(n) where k(n) is the least positive integer not leading to a duplicate term in the sequence.

%C Apparently every positive integer appears in the sequence.

%H Rémy Sigrist, <a href="/A304971/b304971.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A304971/a304971.png">Scatterplot of (n, a(n)) for n = 1..10000000</a>

%F a(n) = (3*a(2*n) - a(2*n+1)) / 2.

%e The first terms, alongside k(n) and associate children, are:

%e n a(n) k(n) a(2*n) a(2*n+1)

%e -- ---- ---- ------ --------

%e 1 1 1 2 4

%e 2 2 1 3 5

%e 3 4 2 6 10

%e 4 3 4 7 15

%e 5 5 3 8 14

%e 6 6 5 11 21

%e 7 10 2 12 16

%e 8 7 2 9 13

%e 9 15 3 18 24

%e 10 8 9 17 35

%o (PARI) lista(nn) = my (a=[1], s=2^a[1]); for (n=1, ceil(nn/2), for (k=1, oo, if (!bittest(s, a[n]+k) && !bittest(s, a[n]+3*k), a=concat(a, [a[n]+k

%o , a[n]+3*k]); s+=2^(a[n]+k) + 2^(a[n]+3*k); break))); a[1..nn]

%Y This sequence is a variant of A305410.

%K nonn,look,nice

%O 1,2

%A _Rémy Sigrist_, Dec 16 2018