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A301336
a(n) = total number of 1's minus total number of 0's in binary expansions of 0, ..., n.
4
-1, 0, 0, 2, 1, 2, 3, 6, 4, 4, 4, 6, 6, 8, 10, 14, 11, 10, 9, 10, 9, 10, 11, 14, 13, 14, 15, 18, 19, 22, 25, 30, 26, 24, 22, 22, 20, 20, 20, 22, 20, 20, 20, 22, 22, 24, 26, 30, 28, 28, 28, 30, 30, 32, 34, 38, 38, 40, 42, 46, 48, 52, 56, 62, 57, 54, 51, 50, 47, 46, 45, 46, 43, 42, 41, 42
OFFSET
0,4
FORMULA
G.f.: -1/(1 - x) + (1/(1 - x)^2)*Sum_{k>=0} x^(2^k)*(1 - x^(2^k))/(1 + x^(2^k)).
a(n) = A000788(n) - A059015(n).
a(n) = A268289(n) - 1.
a(A000079(n)) = A000295(n).
EXAMPLE
+---+-----+---+---+---+---+------------+
| n | bin.|1's|sum|0's|sum| a(n) |
+---+-----+---+---+---+---+------------+
| 0 | 0 | 0 | 0 | 1 | 1 | 0 - 1 =-1 |
| 1 | 1 | 1 | 1 | 0 | 1 | 1 - 1 = 0 |
| 2 | 10 | 1 | 2 | 1 | 2 | 2 - 2 = 0 |
| 3 | 11 | 2 | 4 | 0 | 2 | 4 - 2 = 2 |
| 4 | 100 | 1 | 5 | 2 | 4 | 5 - 4 = 1 |
| 5 | 101 | 2 | 7 | 1 | 5 | 7 - 5 = 2 |
| 6 | 110 | 2 | 9 | 1 | 6 | 9 - 6 = 3 |
+---+-----+---+---+---+---+------------+
bin. - n written in base 2;
1's - number of 1's in binary expansion of n;
0's - number of 0's in binary expansion of n;
sum - total number of 1's (or 0's) in binary expansions of 0, ..., n.
MATHEMATICA
Accumulate[DigitCount[Range[0, 75], 2, 1] - DigitCount[Range[0, 75], 2, 0]]
PROG
(Python)
def A301336(n):
return sum(2*bin(i).count('1')-len(bin(i))+2 for i in range(n+1)) # Chai Wah Wu, Sep 03 2020
KEYWORD
sign,base
AUTHOR
Ilya Gutkovskiy, Mar 28 2018
STATUS
approved