A290256 a(n) is the number of parts equal to 1 in the integer partition having viabin number n.

1, 2, 0, 3, 1, 0, 0, 4, 2, 1, 1, 0, 0, 0, 0, 5, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 6, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Emeric Deutsch, Sep 03 2017

Keywords

• nonn

Comments

The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20.

a(n) is the number of 0's following directly the first 1 in the binary representation of 2n.

Examples

			a(9) = 2. Indeed, the binary form of 9 is 1001; with an additional 0 at the end, it leads to the path ENNEN, where E=(1,0), N=(0,1); this path is the southeast border of the Ferrers board of the integer partition [2,1,1], having 2 parts equal to 1.
		

Crossrefs

Cf. A290255.

Programs

Maple

b := proc(n) if type(log[2](n), integer) then log[2](n) else b(floor((1/2)*n)) end if end proc: seq(b(2*n), n = 1 .. 78);

Formula

a(n) = A290255(2n).