A292342 Number of singletons in the integer partition having viabin number n.

1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 2, 0, 2, 0, 1, 0, 1, 0, 1, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 2, 2, 3, 1, 3, 1, 2, 0, 1, 0, 1, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 2, 0, 2, 0, 1, 1, 2, 1, 2, 2, 4, 2, 3, 1, 2, 2, 3, 1

Offset

1,5

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.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Formula

a(n) = A290260(2n).

Example

a(37) = 2; indeed, the binary representation of 37 is 100101, leading to the integer partition [3',2',1,1] (the singletons are marked).

Maple

a := proc (n) local b: b := proc (n) if n = 1 then 0 elif `mod`(n, 2) = 0 and `mod`((1/2)*n, 2) = 1 then 1+b((1/2)*n) elif `mod`(n, 2) = 1 then b((1/2)*n-1/2) elif `mod`(n-4, 8) = 0 then b((1/2)*n)-1 else b((1/2)*n) end if end proc: b(2*n) end proc: seq(a(n), n = 1 .. 150);

Mathematica

A292342[n_] := Count[Split[IntegerDigits[2*n, 2]], {0}];
Array[A292342, 100] (* Paolo Xausa, Oct 01 2024 *)

Crossrefs

Bisection of A290260.

Sequence in context: A342656 A293896 A066416 * A091991 A108234 A324572

Adjacent sequences: A292339 A292340 A292341 * A292343 A292344 A292345

Keyword

nonn

Author

Emeric Deutsch, Sep 16 2017

Status

approved