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 }.
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
Offset: 0
Examples
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.
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..8192
Programs
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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); }
Comments