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A293664 For n >= 0, let E_n be the set of exponents in expression of 2*n as a sum of distinct powers of 2 (2*n = Sum_{e in E_n} 2^e); a(n) = number of distinct values taken by the expression Sum_{e in E_n} s(e)*2^e when s runs over all functions from the positive numbers to the set { +1, -1 }. 2
1, 2, 2, 4, 2, 4, 4, 7, 2, 4, 4, 8, 4, 7, 8, 11, 2, 4, 4, 8, 4, 8, 7, 12, 4, 7, 8, 13, 8, 12, 13, 16, 2, 4, 4, 8, 4, 8, 8, 13, 4, 8, 7, 14, 8, 13, 14, 17, 4, 7, 8, 13, 8, 14, 13, 18, 8, 12, 14, 19, 15, 18, 19, 22, 2, 4, 4, 8, 4, 8, 8, 14, 4, 8, 8, 15, 7, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

More informally, any number n encodes a finite sets of positive numbers, say { e_1, e_2, ..., e_h }, and a(n) gives the number of distinct values of the form +- e_1 +- e_2 ... +- e_h.

The set of exponents in expression for n as a sum of distinct powers of 2 corresponds to the n-th row of A133457.

A number n belongs to A293576 iff a(n) is odd.

a(n) <= 2^A000120(n) for any n >= 0.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..8192

EXAMPLE

For n = 15:

- E_15 = { 1, 2, 3, 4 },

- the possible "plus-minus" sums are:

  +4 +3 +2 +1 =  10   (1st value)

  +4 +3 +2 -1 =   8   (2nd value)

  +4 +3 -2 +1 =   6   (3rd value)

  +4 +3 -2 -1 =   4   (4th value)

  +4 -3 +2 +1 =   4   (already seen)

  +4 -3 +2 -1 =   2   (5th value)

  +4 -3 -2 +1 =   0   (6th value)

  +4 -3 -2 -1 =  -2   (7th value)

  -4 +3 +2 +1 =   2   (already seen)

  -4 +3 +2 -1 =   0   (already seen)

  -4 +3 -2 +1 =  -2   (already seen)

  -4 +3 -2 -1 =  -4   (8th value)

  -4 -3 +2 +1 =  -4   (already seen)

  -4 -3 +2 -1 =  -6   (9th value)

  -4 -3 -2 +1 =  -8   (10th value)

  -4 -3 -2 -1 = -10   (11th value)

- hence, a(15) = 11.

PROG

(PARI) a(n) = { my (v=Set(0)); my (b = Vecrev(binary(n))); for (i=1, #b, if (b[i], v = setunion(Set(vector(#v, k, v[k]-i)), Set(vector(#v, k, v[k]+i))); ); ); return (#v); }

CROSSREFS

Cf. A133457, A293576.

Sequence in context: A094269 A157227 A054536 * A001316 A285741 A161831

Adjacent sequences:  A293661 A293662 A293663 * A293665 A293666 A293667

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Oct 14 2017

STATUS

approved

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Last modified February 24 06:13 EST 2018. Contains 299597 sequences. (Running on oeis4.)