%I #21 Feb 16 2019 12:12:20
%S 0,1,2,2,3,4,4,5,6,7,8,9,9,10,11,11,12,13,14,15,16,16,17,18,18,19,20,
%T 20,21,22,22,23,24,24,25,26,27,28,29,29,30,31,31,32,33,34,35,36,36,37,
%U 38,38,39,40,40,41,42,42,43,44,44,45,46,47,48,49,49
%N Partial sums of A014578.
%C Duplicate of A050294? [_Joerg Arndt_, Apr 27 2013]
%C From _Michel Dekking_, Feb 10 2019: (Start)
%C The answer to Joerg Arndt's question is: yes (modulo an offset). To see this, it suffices to prove that the two sequences of first differences Da and Db of a= A051068 and b:=A050294 are equal. Clearly the sequence Da of first differences of a is the sequence A014578. According to Philippe Deleham (2004), Da equals 0x = 0110110111110..., where x is the fixed point of the morphism 0->111, 1->110.
%C From _Vladimir Shevelev_ (2011) we know a formula for b=A050294: b(n) = n-b(floor(n/3)). This gives that the sequence of first differences Db:=(b(n+1)-b(n)) of b satisfies
%C Db(3m+1) = Db(3m+2) = 1, and Db(3m+3) = 1 - Db(m).
%C This implies that Db = x, the fixed point of 0->111, 1->110.
%C (End)
%F a(3^n) = A015518(n+1) = -(-1)^n*A014983(n+1). - _Philippe Deléham_, Mar 31 2004
%Y Cf. A014578, A051069, A050294.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, _Gary W. Adamson_