

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
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OFFSET

0,4


LINKS

Table of n, a(n) for n=0..75.
Index entries for sequences related to binary expansion of n


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'ssum0'ssum 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


CROSSREFS

Cf. A000079, A000120, A000295, A000788, A001855, A023416, A037861, A059015, A083652, A145037, A268289, A301896.
Sequence in context: A156564 A198123 A106576 * A128474 A108618 A097719
Adjacent sequences: A301333 A301334 A301335 * A301337 A301338 A301339


KEYWORD

sign,base


AUTHOR

Ilya Gutkovskiy, Mar 28 2018


STATUS

approved



