cp's OEIS Frontend

Duplicate of A050294? [Joerg Arndt, Apr 27 2013]

From Michel Dekking, Feb 10 2019: (Start)

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.

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

Db(3m+1) = Db(3m+2) = 1, and Db(3m+3) = 1 - Db(m).

This implies that Db = x, the fixed point of 0->111, 1->110.

(End)

A050294 is different from this sequence. A050294 involves sets encompassing no {x,3x}; this sequence involves sets encompassing no {x,2x,3x}.

From Steven Finch, Feb 27 2009: (Start)

Define d(n)=A003586(n), b(0)=0 and b(k)=A057561(n) for d(n) <= k < d(n+1).

Then a(n) = Sum_{m=1..ceiling(n/3)} b(floor(n/e(m))) where e(m) = A007310(m). (End)