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a(n+1) is the number of times the binary representation of a(n) is the concatenation of that of one or more consecutive terms in this sequence so far with a(1) = 1.
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%I #14 Jul 10 2020 03:43:52

%S 1,1,2,1,3,2,2,3,3,4,1,4,2,4,3,5,2,5,3,6,2,6,3,7,2,7,3,8,1,5,4,4,5,5,

%T 6,4,6,5,7,4,7,5,8,2,8,3,9,2,9,3,10,2,10,3,11,2,11,3,12,2,12,3,13,3,

%U 14,3,15,2,13,4,8,4,9,4,10,4,11,4,12,4,13,5

%N a(n+1) is the number of times the binary representation of a(n) is the concatenation of that of one or more consecutive terms in this sequence so far with a(1) = 1.

%C This sequence has similarities with A331614 and A332518; here we concatenate consecutive terms, there we add and multiply them, respectively.

%C This sequence is unbounded.

%H Rémy Sigrist, <a href="/A336037/b336037.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A336037/a336037.png">Density plot of the first 100000000 terms</a>

%H Rémy Sigrist, <a href="/A336037/a336037.gp.txt">PARI program for A336037</a>

%e The first terms, alongside their binary representation and the corresponding concatenations, are:

%e n a(n) bin(a(n)) Concatenations

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

%e 1 1 1 a(1)

%e 2 1 1 a(1), a(2)

%e 3 2 10 a(3)

%e 4 1 1 a(1), a(2), a(4)

%e 5 3 11 a(1)|a(2), a(5)

%e 6 2 10 a(3), a(6)

%e 7 2 10 a(3), a(6), a(7)

%e 8 3 11 a(1)|a(2), a(5), a(8)

%e 9 3 11 a(1)|a(2), a(5), a(8), a(9)

%e 10 4 100 a(10)

%o (PARI) See Links section.

%Y Cf. A331614, A332518.

%K nonn,base

%O 1,3

%A _Rémy Sigrist_, Jul 07 2020