A051068 Partial sums of A014578.

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, 20, 21, 22, 22, 23, 24, 24, 25, 26, 27, 28, 29, 29, 30, 31, 31, 32, 33, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 47, 48, 49, 49

Offset: 0

Author

N. J. A. Sloane, Gary W. Adamson

Keywords

• nonn

Comments

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)

Crossrefs

Cf. A014578, A051069, A050294.

Formula

a(3^n) = A015518(n+1) = -(-1)^n*A014983(n+1). - Philippe Deléham, Mar 31 2004