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A290260 a(n) = number of isolated 0's in the binary representation of n. 2
0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 2, 2, 1, 0, 0, 2, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 3, 2, 1, 2, 2, 1, 0, 0, 1, 0, 1, 2, 2, 1, 0, 0, 2, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 3, 3, 2, 1, 1, 3, 2, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 3, 2, 1, 2, 2, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,10

COMMENTS

a(2n) = number of singletons in the integer partition having viabin number n. 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 [5,4,4,3,1]. The southeast border of its Ferrers board yields 101101001, leading to the viabin number 361.

The following 5 formulae are used for the Maple program.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..65536

FORMULA

a(1) = 0.

If n is odd, then a(2n) = 1 + a(n).

a(2n+1) = a(n).

If n-2 mod 4 = 0, then a(2n) = a(n) - 1.

If n mod 8 = 0 then a(2n) = a(n).

EXAMPLE

a(722) = 3; indeed, the binary representation of 722 is 1011010010, having 3 isolated 0's.

MAPLE

a := proc (n) if n = 1 then 0 elif `mod`(n, 2) = 0 and `mod`((1/2)*n, 2) = 1 then 1+a((1/2)*n) elif `mod`(n, 2) = 1 then a((1/2)*n-1/2) elif `mod`(n-4, 8) = 0 then a((1/2)*n)-1 else a((1/2)*n) end if end proc: seq(a(n), n = 1 .. 200);

# second Maple program:

b:= n-> `if`(n<6, 0, b(iquo(n, 2)))+`if`(n mod 8=5, 1, 0):

a:= n-> b(2*n+1):

seq(a(n), n=1..200);  # Alois P. Heinz, Sep 14 2017

MATHEMATICA

Table[Count[Split@ IntegerDigits[n, 2], {0}], {n, 105}] (* Michael De Vlieger, Sep 16 2017 *)

CROSSREFS

Cf. A292342.

Sequence in context: A208134 A175560 A143240 * A304273 A153659 A305565

Adjacent sequences:  A290257 A290258 A290259 * A290261 A290262 A290263

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 14 2017

STATUS

approved

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Last modified September 21 07:28 EDT 2021. Contains 347596 sequences. (Running on oeis4.)