A023416 -id:A023416 - cp's OEIS frontend

A031443 Digitally balanced numbers: positive numbers that in base 2 have the same number of 0's as 1's.

2, 9, 10, 12, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 177, 178, 180, 184, 195, 197, 198, 201, 202, 204, 209, 210, 212, 216, 225, 226, 228, 232, 240, 527, 535, 539, 541, 542, 551

Offset: 1

Clark Kimberling

Also numbers k such that the binary digital mean dm(2, k) = (Sum_{i=1..d} 2*d_i - 1) / (2*d) = 0, where d is the number of digits in the binary representation of k and d_i the individual digits. - Reikku Kulon, Sep 21 2008
From Reikku Kulon, Sep 29 2008: (Start)
Each run of values begins with 2^(2k + 1) + 2^(k + 1) - 2^k - 1. The initial values increase according to the sequence {2^(k - 1), 2^(k - 2), 2^(k - 3), ..., 2^(k - k)}.
After this, the values follow a periodic sequence of increases by successive powers of two with single odd values interspersed.
Each run ends with an odd increase followed by increases of {2^(k - k), ..., 2^(k - 2), 2^(k - 1), 2^k}, finally reaching 2^(2k + 2) - 2^(k + 1).
Similar behavior occurs in other bases. (End)
Numbers k such that A000120(k)/A070939(k) = 1/2. - Ctibor O. Zizka, Oct 15 2008
Subsequence of A053754; A179888 is a subsequence. - Reinhard Zumkeller, Jul 31 2010
A000120(a(n)) = A023416(a(n)); A037861(a(n)) = 0.
A001700 gives number of terms having length 2*n in binary representation: A001700(n-1) = #{m: A070939(a(m))=2*n}. - Reinhard Zumkeller, Jun 08 2011
The number of terms below 2^k is A079309(floor(k/2)) for k > 1. - Amiram Eldar, Nov 21 2020

			9 is a term because '1001' contains 2 '0's and 2 '1's.

Reikku Kulon, Table of n, a(n) for n = 1..10000
Jason Bell, Thomas F. Lidbetter and Jeffrey Shallit, Additive number theory via approximation by regular languages, International Journal of Foundations of Computer Science, Vol. 31, No. 6 (2020), pp. 667-687; arXiv preprint, arXiv:1804.07996 [cs.FL], 2018.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Reinhard Zumkeller, Haskell Programs for Binary Digitally Balanced Numbers.
Index entries for sequences related to binary expansion of n

Subsequences: A144798, A144799, A144800, A144801, A144812, A179888.
Subsequence of A053754.
Row n = 2 of A378000.
Cf. A049354-A049360, A000120, A001316, A006519, A023416, A070939, A144777, A145057, A145058, A145059, A145060, A144912, A145037, A191292, A090050, A014486, A061854, A037861, A079309.
Terms of binary width n are enumerated by A001700.

-- See link, showing that Ulrich Schimkes formula provides a very efficient algorithm. Reinhard Zumkeller, Jun 15 2011

[ n: n in [2..250] | Multiplicity({* z: z in Intseq(n,2) *}, 0) eq &+Intseq(n,2) ];  // Bruno Berselli, Jun 07 2011

a:=proc(n) local nn, n1, n0: nn:=convert(n,base,2): n1:=add(nn[i],i=1..nops(nn)): n0:=nops(nn)-n1: if n0=n1 then n else end if end proc: seq(a(n), n = 1..240); # Emeric Deutsch, Jul 31 2008

Select[Range[250],DigitCount[#,2,1]==DigitCount[#,2,0]&] (* Harvey P. Dale, Jul 22 2013 *)
FromDigits[#,2]&/@DeleteCases[Flatten[Permutations/@Table[PadRight[{},2n,{1,0}],{n,5}],1],?(#[[1]]==0&)]//Sort (* _Harvey P. Dale, May 30 2016 *)

for(n=1,100,b=binary(n); l=length(b); if(sum(i=1,l, component(b,i))==l/2,print1(n,",")))

is(n)=hammingweight(n)==hammingweight(bitneg(n,#binary(n))) \\ Charles R Greathouse IV, Mar 29 2013

is(n)=2*hammingweight(n)==exponent(n)+1 \\ Charles R Greathouse IV, Apr 18 2020

for my $half ( 1 .. 4 ) {
  my $N = 2 * $half;  # only even widths apply
  my $vector = (1 << ($N-1)) | ((1 << ($N/2-1)) - 1);  # first key
  my $n = 1; $n *= $_ for 2 .. $N;    # N!
  my $d = 1; $d *= $_ for 2 .. $N/2;  # (N/2)!
  for (1 .. $n/($d*$d*2)) {
    print "$vector, ";
    my ($v, $d) = ($vector, 0);
    until ($v & 1 or !$v) { $d = ($d << 1)|1; $v >>= 1 }
    $vector += $d + 1 + (($v ^ ($v + 1)) >> 2);  # next key
  }
} # Ruud H.G. van Tol, Mar 30 2014

from sympy.utilities.iterables import multiset_permutations
A031443_list = [int('1'+''.join(p),2) for n in range(1,10) for p in multiset_permutations('0'*n+'1'*(n-1))] # Chai Wah Wu, Nov 15 2019

a(n+1) = a(n) + 2^k + 2^(m-1) - 1 + floor((2^(k+m) - 2^k)/a(n))*(2^(2*m) + 2^(m-1)) where k is the largest integer such that 2^k divides a(n) and m is the largest integer such that 2^m divides a(n)/2^k+1. - Ulrich Schimke (UlrSchimke(AT)aol.com)

A145037(a(n)) = 0. - Reikku Kulon, Oct 02 2008

A080791 Number of nonleading 0's in binary expansion of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 0, 3, 2, 2, 1, 2, 1, 1, 0, 4, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 0, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 4, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 0, 6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3, 2, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 5, 4, 4, 3, 4, 3, 3, 2, 4

Offset: 0

Cino Hilliard, Mar 25 2003

In this version we consider the number zero to have no nonleading 0's, thus a(0) = 0. The variant A023416 has a(0) = 1.

Number of steps required to reach 1, starting at n + 1, under the operation: if x is even divide by 2 else add 1. This is the x + 1 problem (as opposed to the 3x + 1 problem).

			a(4) = 2 since 4 in binary is 100, which has two zeros.
a(5) = 1 since 5 in binary is 101, which has only one zero.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A000120, A007814, A023416, A029837, A036987, A080791-A080801, A048881, A054429, A092339, A102364, A120511, A233271, A233272, A233273.

seq(numboccur(0, Bits[Split](n)), n=0..100); # Robert Israel, Oct 26 2017

{0}~Join~Table[Last@ DigitCount[n, 2], {n, 120}] (* Michael De Vlieger, Mar 07 2016 *)
f[n_] := If[OddQ@ n, f[n -1] -1, f[n/2] +1]; f[0] = f[1] = 0; Array[f, 105, 0] (* Robert G. Wilson v, May 21 2017 *)
Join[{0}, Table[Count[IntegerDigits[n, 2], 0], {n, 1, 100}]] (* Vincenzo Librandi, Oct 27 2017 *)

PARI
```
a(n)=if(n,a(n\2)+1-n%2)
```

A080791(n)=if(n,logint(n,2)+1-hammingweight(n)) \\ M. F. Hasler, Oct 26 2017

def a(n): return bin(n)[2:].count("0") if n>0 else 0 # Indranil Ghosh, Apr 10 2017

;; with memoizing definec-macro from Antti Karttunen's IntSeq-library)
(define (A080791 n) (- (A029837 (+ 1 n)) (A000120 n)))
;; Alternative version based on a simple recurrence:
(definec (A080791 n) (if (zero? n) 0 (+ (A080791 (- n 1)) (A007814 n) (A036987 (- n 1)) -1)))
;; from Antti Karttunen, Dec 12 2013

From Antti Karttunen, Dec 12 2013: (Start)
a(n) = A029837(n+1) - A000120(n).
a(0) = 0, and for n > 0, a(n) = (a(n-1) + A007814(n) + A036987(n-1)) - 1.
For all n >= 1, a(A054429(n)) = A048881(n-1) = A000120(n) - 1.
Equally, for all n >= 1, a(n) = A000120(A054429(n)) - 1.
(End)
Recurrence: a(2n) = a(n) + 1 (for n > 0), a(2n + 1) = a(n). - Ralf Stephan from Cino Hilliard's PARI program, Dec 16 2013. Corrected by Alonso del Arte, May 21 2017 after consultation with Chai Wah Wu and Ray Chandler, "n > 0" added by M. F. Hasler, Oct 26 2017
a(n) = A023416(n) for all n > 0. - M. F. Hasler, Oct 26 2017
G.f. g(x) satisfies g(x) = (1+x)*g(x^2) + x^2/(1-x^2). - Robert Israel, Oct 26 2017

A000788 Total number of 1's in binary expansions of 0, ..., n.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 9, 12, 13, 15, 17, 20, 22, 25, 28, 32, 33, 35, 37, 40, 42, 45, 48, 52, 54, 57, 60, 64, 67, 71, 75, 80, 81, 83, 85, 88, 90, 93, 96, 100, 102, 105, 108, 112, 115, 119, 123, 128, 130, 133, 136, 140, 143, 147, 151, 156, 159, 163, 167, 172, 176, 181, 186

Offset: 0

N. J. A. Sloane

Partial sums of A000120.
The graph of this sequence is a version of the Takagi curve: see Lagarias (2012), Section 9, especially Theorem 9.1. - N. J. A. Sloane, Mar 12 2016
a(n-1) is the largest possible number of ordered pairs (a,b) such that a/b is a prime in a subset of the positive integers with n elements. - Yifan Xie, Feb 21 2025

J.-P. Allouche & J. Shallit, Automatic sequences, Cambridge University Press, 2003, p. 94
R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340. See Eq. 1.9. [From N. J. A. Sloane, Mar 12 2009]
L. E. Bush, An asymptotic formula for the average sums of the digits of integers, Amer. Math. Monthly, 47 (1940), pp. 154-156. [From the bibliography of Stolarsky, 1977]
P. Cheo and S. Yien, A problem on the k-adic representation of positive integers (Chinese; English summary), Acta Math. Sinica, 5 (1955), pp. 433-438. [From the bibliography of Stolarsky, 1977]
M. P. Drazin and J. S. Griffith, On the decimal representation of integers, Proc. Cambridge Philos. Soc., (4), 48 (1952), pp. 555-565. [From the bibliography of Stolarsky, 1977]
E. N. Gilbert, Games of identification or convergence, SIAM Review, 4 (1962), 16-24.
Grabner, P. J.; Kirschenhofer, P.; Prodinger, H.; Tichy, R. F.; On the moments of the sum-of-digits function. Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
R. L. Graham, On primitive graphs and optimal vertex assignments, pp. 170-186 of Internat. Conf. Combin. Math. (New York, 1970), Annals of the NY Academy of Sciences, Vol. 175, 1970.
E. Grosswald, Properties of some arithmetic functions, J. Math. Anal. Appl., 28 (1969), pp.405-430.
Donald E. Knuth, The Art of Computer Programming, volume 3 Sorting and Searching, section 5.3.4, subsection Bitonic sorting, with C'(p) = a(p-1).
Hiu-Fai Law, Spanning tree congestion of the hypercube, Discrete Math., 309 (2009), 6644-6648 (see p(m) on page 6647).
Z. Li and E. M. Reingold, Solution of a divide-and-conquer maximin recurrence, SIAM J. Comput., 18 (1989), 1188-1200.
B. Lindström, On a combinatorial problem in number theory, Canad. Math. Bull., 8 (1965), 477-490.
Mauclaire, J.-L.; Murata, Leo; On q-additive functions. I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
Mauclaire, J.-L.; Murata, Leo; On q-additive functions. II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
M. D. McIlroy, The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput., 3 (1974), 255-261.
L. Mirsky, A theorem on representations of integers in the scale of r, Scripta Math., 15 (1949), pp. 11-12.
I. Shiokawa, On a problem in additive number theory, Math. J. Okayama Univ., 16 (1974), pp.167-176. [From the bibliography of Stolarsky, 1977]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
K. B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math., 32 (1977), 717-730.
Trollope, J. R. An explicit expression for binary digital sums. Math. Mag. 41 1968 21-25.

T. D. Noe and Hieronymus Fischer, Table of n, a(n) for n = 0..10000 (terms up to n=1000 by T. D. Noe).
G. Agnarsson, On the number of hypercubic bipartitions of an integer, arXiv preprint arXiv:1106.4997 [math.CO], 2011.
G. Agnarsson, Induced subgraphs of hypercubes, arXiv preprint arXiv:1112.3015 [math.CO], 2011.
G. Agnarsson and K. Lauria, Extremal subgraphs of the d-dimensional grid graph, arXiv preprint arXiv:1302.6517 [math.CO], 2013.
J.-P. Allouche, On an Inequality in a 1970 Paper of R. L. Graham, INTEGERS 21A (2021), #A2.
Mathias Hauan Arbo, Esten Ingar Grøtli, and Jan Tommy Gravdahl, CASCLIK: CasADi-Based Closed-Loop Inverse Kinematics, arXiv:1901.06713 [cs.RO], 2019.
Venkata Sai Narayana Bavisetty, Matthew Wheeler, Reinhard Laubenbacher, and Claus Kadelka, Upper bound for the stability of Boolean networks, arXiv:2506.12310 [q-bio.MN], 2025. See p. 8.
Johann Cigler, A curious class of Hankel determinants, arXiv:1803.05164 [math.CO], 2018.
G. F. Clements and B. Lindström, A sequence of (+-1) determinants with large values, Proc. Amer. Math. Soc., 16 (1965), pp. 548-550. [From the bibliography of Stolarsky, 1977]
Jean Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
H. Delange, Sur la fonction sommatoire de la fonction "somme des chiffres", Enseignement Math., (2), 21 (1975), pp. 31-47. [From the bibliography of Stolarsky, 1977]
Laurent Feuilloley, Brief Announcement: Average Complexity for the LOCAL Model, arXiv preprint arXiv:1505.05072 [cs.DC], 2015.
S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.
Oscar E. González, An observation of Rankin on Hankel determinants, Department of Mathematics, University of Illinois at Urbana-Champaign, 2018.
P. J. Grabner and H.-K. Hwang, Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence
Milton W. Green, Letter to N. J. A. Sloane, 1973 (note "A360" refers to N0360 which is the present sequence).
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 36 and 44.
Jeffrey C. Lagarias, The Takagi function and its properties, arXiv:1112.4205 [math.CA], 2011-2012.
Jeffrey C. Lagarias, The Takagi function and its properties, In Functions in number theory and their probabilistic aspects, 153--189, RIMS Kôkyûroku Bessatsu, B34, Res. Inst. Math. Sci. (RIMS), Kyoto, 2012. MR3014845.
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Raoul Nakhmanson-Kulish, Graphic representation of K(n)=a(n)/(n log2(n)/2) from n=63 to 131071
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Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
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Eric Weisstein's World of Mathematics, Binary
David W. Wilson, Fast C++ function for computing a(n)
Index entries for sequences related to binary expansion of n

For number of 0's in binary expansion of 0, ..., n see A059015.
The basic sequences concerning the binary expansion of n are A000120, A000788, A000069, A001969, A023416, A059015, A070939, A083652.
Cf. A005183, A360189.
Cf. A027868, A037123, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564 (for base 10).

a000788_list = scanl1 (+) A000120_list
-- Walt Rorie-Baety, Jun 30 2012

{a000788 0 = 0; a00788 n = a000788 n2 + a000788 (n-n2-1) + (n-n2) where n2 = n `div` 2}
-- Walt Rorie-Baety, Jul 15 2012

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+add(i, i=Bits[Split](n))) end:
seq(a(n), n=0..62);  # Alois P. Heinz, Nov 11 2024

a[n_] := Count[ Table[ IntegerDigits[k, 2], {k, 0, n}], 1, 2]; Table[a[n], {n, 0, 62}] (* Jean-François Alcover, Dec 16 2011 *)
Table[Plus@@Flatten[IntegerDigits[Range[n], 2]], {n, 0, 62}] (* Alonso del Arte, Dec 16 2011 *)
Accumulate[DigitCount[Range[0,70],2,1]] (* Harvey P. Dale, Jun 08 2013 *)

A000788(n)={ n<3 && return(n); if( bittest(n,0) \\
, n+1 == 1<A000788(n>>1)*2+n>>1+1 \\
, n == 1<A000788(n>>=1)+A000788(n-1)+n )} \\ M. F. Hasler, Nov 22 2009

a(n)=sum(k=1,n,hammingweight(k)) \\ Charles R Greathouse IV, Oct 04 2013

a(n) = if (n==0, 0, m = logint(n, 2); r = n % 2^m; m*2^(m-1) + r + 1 + a(r)); \\ Michel Marcus, Mar 27 2018

a(n)={n++; my(t, i, s); c=n; while(c!=0, i++; c\=2); for(j=1, i, d=(n\2^(i-j))%2; t+=(2^(i-j)*(s*d+d*(i-j)/2)); s+=d); t} \\ David A. Corneth, Nov 26 2024
(C++) /* See David W. Wilson link. */

def A000788(n): return sum(i.bit_count() for i in range(1,n+1)) # Chai Wah Wu, Mar 01 2023

def A000788(n): return (n+1)*n.bit_count()+(sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1))>>1) # Chai Wah Wu, Nov 11 2024

McIlroy (1974) gives bounds and recurrences. - N. J. A. Sloane, Mar 24 2014
Stolarsky (1977) studies the asymptotics, and gives at least nine references to earlier work on the problem. I have added all the references that were not here already. - N. J. A. Sloane, Apr 06 2014
a(n) = Sum_{k=1..n} A000120(k). - Benoit Cloitre, Dec 19 2002
a(0) = 0, a(2n) = a(n)+a(n-1)+n, a(2n+1) = 2a(n)+n+1. - Ralf Stephan, Sep 13 2003
a(n) = n*log_2(n)/2 + O(n); a(2^n)=n*2^(n-1)+1. - Benoit Cloitre, Sep 25 2003 (The first result is due to Bellman and Shapiro, - N. J. A. Sloane, Mar 24 2014)
a(n) = n*log_2(n)/2+n*F(log_2(n)) where F is a nowhere differentiable continuous function of period 1 (see Allouche & Shallit). - Benoit Cloitre, Jun 08 2004
G.f.: (1/(1-x)^2) * Sum_{k>=0} x^2^k/(1+x^2^k). - Ralf Stephan, Apr 19 2003
a(2^n-1) = A001787(n) = n*2^(n-1). - M. F. Hasler, Nov 22 2009
a(4^n-2) = n(4^n-2).
For real n, let f(n) = [n]/2 if [n] even, n-[n+1]/2 otherwise. Then a(n) = Sum_{k>=0} 2^k*f((n+1)/2^k).
a(A000225(n)) = A173921(A000225(n)) = A001787(n); a(A000079(n)) = A005183(n). - Reinhard Zumkeller, Mar 04 2010
From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/2^j + 1/2)*(2n + 2 - floor(n/2^j + 1/2))*2^j - floor(n/2^j)*(2n + 2 - (1 + floor(n/2^j)) * 2^j)), where m=floor(log_2(n)).
a(n) = (n+1)*A000120(n) - 2^(m-1) + 1/4 + (1/2)*Sum_{j=1..m+1} ((floor(n/2^j) + 1/2)^2 - floor(n/2^j + 1/2)^2)*2^j, where m=floor(log_2(n)).
a(2^m-1) = m*2^(m-1).
(This is the total number of '1' digits occurring in all the numbers with <= m bits.)
Generic formulas for the number of digits >= d in the base p representations of all integers from 0 to n, where 1<= d < p.
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/p^j + (p-d)/p)*(2n + 2 + ((p-2*d)/p - floor(n/p^j + (p-d)/p))*p^j) - floor(n/p^j)*(2n + 2 - (1+floor(n/p^j)) * p^j)), where m=floor(log_p(n)).
a(n) = (n+1)*F(n,p,d) + (1/2)*Sum_{j=1..m+1} ((((p-2*d)/p)*floor(n/p^j+(p-d)/p) + floor(n/p^j))*p^j - (floor(n/p^j+(p-d)/p)^2 - floor(n/p^j)^2)*p^j), where m=floor(log_p(n)) and F(n,p,d) = number of digits >= d in the base p representation of n.
a(p^m-1) = (p-d)*m*p^(m-1).
(This is the total number of digits >= d occurring in all the numbers  with <= m digits in base p representation.)
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(d*p^j) - x^(p*p^j))/(1-x^(p*p^j)). (End)
a(n) = Sum_{k=1..n} A000120(A240857(n,k)). - Reinhard Zumkeller, Apr 14 2014
For n > 0, if n is written as 2^m + r with 0 <= r < 2^m, then a(n) = m*2^(m-1) + r + 1 + a(r). - Shreevatsa R, Mar 20 2018
a(n) = n*(n+1)/2 + Sum_{k=1..floor(n/2)} ((2k-1)((g(n,k)-1)*2^(g(n,k) + 1) + 2) - (n+1)*(g(n,k)+1)*g(n,k)/2), where g(n,k) = floor(log_2(n/(2k-1))). - Fabio Visonà, Mar 17 2020
From Jeffrey Shallit, Aug 07 2021: (Start)
A 2-regular sequence, satisfying the identities
a(4n+1) = -a(2n) + a(2n+1) + a(4n)
a(4n+2) = -2a(2n) + 2a(2n+1) + a(4n)
a(4n+3) = -4a(n) + 4a(2n+1)
a(8n) = 4a(n) - 8a(2n) + 5a(4n)
a(8n+4) = -9a(2n) + 5a(2n+1) + 4a(4n)
for n>=0. (End)
a(n) = Sum_{k=0..floor(log_2(n+1))} k * A360189(n,k). - Alois P. Heinz, Mar 06 2023

More terms from Larry Reeves (larryr(AT)acm.org), Jan 15 2001

A055641 Number of zero digits in n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 1, 1

Offset: 0

Henry Bottomley, Jun 06 2000

			a(99) = 0 because the digits of 99 are 9 and 9, a(100) = 2 because the digits of 100 are 1, 0 and 0 and there are two 0's.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A011540, A004719, A052382, A054899, A055640, A102669-A102685, A122840, A160093, A160094, A196563, A195564, A000120, A000788, A023416, A059015 (for base 2), A085974.

a055641 n | n < 10    = 0 ^ n
          | otherwise = a055641 n' + 0 ^ d where (n',d) = divMod n 10
-- Reinhard Zumkeller, Apr 30 2013

Array[Last@ DigitCount@ # &, 105] (* Michael De Vlieger, Jul 02 2015 *)

a(n)=if(n,n=digits(n); sum(i=2,#n,n[i]==0), 1) \\ Charles R Greathouse IV, Sep 13 2015

A055641(n)=#select(d->!d,digits(n))+!n \\ M. F. Hasler, Jun 22 2018

def a(n): return str(n).count("0")
print([a(n) for n in range(106)]) # Michael S. Branicky, May 26 2022

From Hieronymus Fischer, Jun 06 2012: (Start)
a(n) = m + 1 - A055640(n) = Sum_{j=1..m+1} (1 + floor(n/10^j) - floor(n/10^j+0.9)), where m = floor(log_10(n)).
G.f.: g(x) = 1 + (1/(1-x))*Sum_{j>=0} (x^(10*10^j) - x^(11*10^j))/(1-x^10^(j+1)). (End)
a(n) = if n<10 then A000007(n) else a(A059995(n)) + A000007(A010879(n)). - Reinhard Zumkeller, Apr 30 2013, corrected by M. F. Hasler, Jun 22 2018

A003754 Numbers with no adjacent 0's in binary expansion.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 23, 26, 27, 29, 30, 31, 42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 63, 85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 127, 170, 171, 173, 174, 175, 181

Offset: 1

N. J. A. Sloane

Theorem (J.-P. Allouche, J. Shallit, G. Skordev): This sequence = A052499 - 1.
Ahnentafel numbers of ancestors contributing the X-chromosome to a female. A280873 gives the male inheritance. - Floris Strijbos, Jan 09 2017 [Equivalence with this sequence pointed out by John Blythe Dobson, May 09 2018]
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. This sequence lists all numbers k such that the k-th composition in standard order has no parts greater than two. See the corresponding example below. - Gus Wiseman, Apr 04 2020
The binary representation of a(n+1) has the same string of digits as the lazy Fibonacci (also known as dual Zeckendorf) representation of n that uses 0s and 1s. (The "+1" is essentially an adjustment for the offset of this sequence.) - Peter Munn, Sep 06 2022

			21 is in the sequence because 21 = 10101_2. '10101' has no '00' present in it. - _Indranil Ghosh_, Feb 11 2017
From _Gus Wiseman_, Apr 04 2020: (Start)
The terms together with the corresponding compositions begin:
    0: ()            30: (1,1,1,2)         90: (2,1,2,2)
    1: (1)           31: (1,1,1,1,1)       91: (2,1,2,1,1)
    2: (2)           42: (2,2,2)           93: (2,1,1,2,1)
    3: (1,1)         43: (2,2,1,1)         94: (2,1,1,1,2)
    5: (2,1)         45: (2,1,2,1)         95: (2,1,1,1,1,1)
    6: (1,2)         46: (2,1,1,2)        106: (1,2,2,2)
    7: (1,1,1)       47: (2,1,1,1,1)      107: (1,2,2,1,1)
   10: (2,2)         53: (1,2,2,1)        109: (1,2,1,2,1)
   11: (2,1,1)       54: (1,2,1,2)        110: (1,2,1,1,2)
   13: (1,2,1)       55: (1,2,1,1,1)      111: (1,2,1,1,1,1)
   14: (1,1,2)       58: (1,1,2,2)        117: (1,1,2,2,1)
   15: (1,1,1,1)     59: (1,1,2,1,1)      118: (1,1,2,1,2)
   21: (2,2,1)       61: (1,1,1,2,1)      119: (1,1,2,1,1,1)
   22: (2,1,2)       62: (1,1,1,1,2)      122: (1,1,1,2,2)
   23: (2,1,1,1)     63: (1,1,1,1,1,1)    123: (1,1,1,2,1,1)
   26: (1,2,2)       85: (2,2,2,1)        125: (1,1,1,1,2,1)
   27: (1,2,1,1)     86: (2,2,1,2)        126: (1,1,1,1,1,2)
   29: (1,1,2,1)     87: (2,2,1,1,1)      127: (1,1,1,1,1,1,1)
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..50000 (terms 1..1000 from T. D. Noe)
J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math., Vol. 292, No. 1-3 (2005), pp. 1-15.
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Tomi Kärki, Anne Lacroix, and Michel Rigo, On the recognizability of self-generating sets, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.2.
Wikipedia, Ahnentafel.
Witzel, Stefan On panel-regular ~A2 lattices Geom. Dedicata 191, 85-135 (2017).
Index entries for 2-automatic sequences.
Index entries for sequences related to binary expansion of n.

A104326(n) = A007088(a(n)); A023416(a(n)) = A087116(a(n)); A107782(a(n)) = 0; A107345(a(n)) = 1; A107359(n) = a(n+1) - a(n); a(A001911(n)) = A000225(n); a(A000071(n+2)) = A000975(n). - Reinhard Zumkeller, May 25 2005
Cf. A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A004742 (no 101), A004743 (no 110), A003726 (no 111).
Complement of A004753.
Cf. A023705, A196168, A280873.
Positions of numbers <= 2 in A333766 (see this and A066099 for other sequences about compositions in standard order).
Cf. A318928.

a003754 n = a003754_list !! (n-1)
a003754_list = filter f [0..] where
   f x = x == 0 || x `mod` 4 > 0 && f (x `div` 2)
-- Reinhard Zumkeller, Dec 07 2012, Oct 19 2011

isA003754 := proc(n) local bdgs ; bdgs := convert(n,base,2) ; for i from 2 to nops(bdgs) do if op(i,bdgs)=0 and op(i-1,bdgs)= 0 then return false; end if; end do; return true; end proc:
A003754 := proc(n) option remember; if n= 1 then 0; else for a from procname(n-1)+1 do if isA003754(a) then return a; end if; end do: end if; end proc:
# R. J. Mathar, Oct 23 2010

Select[ Range[0, 200], !MatchQ[ IntegerDigits[#, 2], {_, 0, 0, _}]&] (* Jean-François Alcover, Oct 25 2011 *)
Select[Range[0,200],SequenceCount[IntegerDigits[#,2],{0,0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, May 21 2015 *)

is(n)=n=bitor(n,n>>1)+1; n>>=valuation(n,2); n==1 \\ Charles R Greathouse IV, Feb 06 2017

i=0
while i<=500:
    if "00" not in bin(i)[2:]:
        print(str(i), end=',')
    i+=1 # Indranil Ghosh, Feb 11 2017

Sum_{n>=2} 1/a(n) = 4.356588498070498826084131338899394678478395568880140707240875371925764128502... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

Removed "2" from the name, because, for example, one could argue that 10001 has 3 adjacent zeros, not 2. - Gus Wiseman, Apr 04 2020

A087207 A binary representation of the primes that divide a number, shown in decimal.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 8, 1, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 3, 4, 33, 2, 9, 512, 7, 1024, 1, 18, 65, 12, 3, 2048, 129, 34, 5, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 3, 20, 9, 130, 513, 65536, 7, 131072, 1025, 10, 1, 36, 19, 262144, 65, 258

Offset: 1

Mitch Cervinka (puritan(AT)planetkc.com), Oct 26 2003

The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.
For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017
From Antti Karttunen, Jun 18 & 20 2017: (Start)
A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n).
For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself.
Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565.
If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612.
(End)
Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024

			a(38) = 129 because 38 = 2*19 = prime(1)*prime(8) and 129 = 2^0 + 2^7 (in binary 10000001).
a(140) = 13, binary 1101 because 140 is divisible by the first, third and fourth primes and 2^(1-1) + 2^(3-1) + 2^(4-1) = 13.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 [First 1000 terms from _T. D. Noe_]
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

For partial sums see A288566.
Cf. A000040, A000120, A001221, A005117, A008479, A019565, A055396, A285320, A285321, A285329, A285330, A285332.
Sequences with related definitions: A007947, A008472, A027748, A048675, A248663, A276379 (same sequence shown in base 2), A288569, A289271, A297404.
Cf. A286608 (numbers n for which a(n) < n), A286609 (n for which a(n) > n), and also A286611, A286612.
A003986, A003961, A059896 are used to express relationship between terms of this sequence.
Related to A267116 via A225546.
Positions of particular values are: A000079\{1} (1), A000244\{1} (2), A033845 (3), A000351\{1} (4), A033846 (5), A033849 (6), A143207 (7), A000420\{1} (8), A033847 (9), A033850 (10), A033851 (12), A147576 (14), A147571 (15), A001020\{1} (16), A033848 (17).
A048675 gives binary rank of prime indices.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices (listed A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- sum A029931, product A096111
- max A029837 or A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000720, A005940, A018819, A023506, A071814, A225620, A277319, A277905, A304818, A372689, A372890.

a087207 = sum . map ((2 ^) . (subtract 1) . a049084) . a027748_row
-- Reinhard Zumkeller, Jul 16 2013

a[n_] := Total[ 2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]; a[1] = 0; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Dec 12 2011 *)

a(n) = {if (n==1, 0, my(f=factor(n), v = []); forprime(p=2, vecmax(f[,1]), v = concat(v, vecsearch(f[,1], p)!=0);); fromdigits(Vecrev(v), 2));} \\ Michel Marcus, Jun 05 2017

A087207(n)=vecsum(apply(p->1<M. F. Hasler, Jun 23 2017

from sympy import factorint, primepi
def a(n):
    return sum(2**primepi(i - 1) for i in factorint(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

(definec (A087207 n) (if (= 1 n) 0 (+ (A000079 (+ -1 (A055396 n))) (A087207 (A028234 n))))) ;; This uses memoization-macro definec
(define (A087207 n) (A048675 (A007947 n))) ;; Needs code from A007947 and A048675. - Antti Karttunen, Jun 19 2017

Additive with a(p^e) = 2^(i-1) where p is the i-th prime. - Vladeta Jovovic, Oct 29 2003
a(n) gives the m such that A019565(m) = A007947(n). - Naohiro Nomoto, Oct 30 2003
A000120(a(n)) = A001221(n); a(n) = Sum(2^(A049084(p)-1): p prime-factor of n). - Reinhard Zumkeller, Nov 30 2003
G.f.: Sum_{k>=1} 2^(k-1)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
From Antti Karttunen, Apr 17 2017, Jun 19 2017 & Dec 06 2018: (Start)
a(n) = A048675(A007947(n)).
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A028234(n)).
A000035(a(n)) = 1 - A000035(n). [a(n) and n are of opposite parity.]
A248663(n) <= a(n) <= A048675(n). [XOR-, OR- and +-variants.]
a(A293214(n)) = A218403(n).
a(A293442(n)) = A267116(n).
A069010(a(n)) = A287170(n).
A007088(a(n)) = A276379(n).
A038374(a(n)) = A300820(n) for n >= 1.
(End)
From Peter Munn, Jan 08 2020: (Start)
a(A059896(n,k)) = a(n) OR a(k) = A003986(a(n), a(k)).
a(A003961(n)) = 2*a(n).
a(n^2) = a(n).
a(n) = A267116(A225546(n)).
a(A225546(n)) = A267116(n).
(End)

More terms from Don Reble, Ray Chandler and Naohiro Nomoto, Oct 28 2003

Name clarified by Antti Karttunen, Jun 18 2017

A059015 Total number of 0's in binary expansions of 0, ..., n.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 6, 6, 9, 11, 13, 14, 16, 17, 18, 18, 22, 25, 28, 30, 33, 35, 37, 38, 41, 43, 45, 46, 48, 49, 50, 50, 55, 59, 63, 66, 70, 73, 76, 78, 82, 85, 88, 90, 93, 95, 97, 98, 102, 105, 108, 110, 113, 115, 117, 118, 121, 123, 125, 126, 128, 129, 130, 130, 136, 141

Offset: 0

Patrick De Geest, Dec 15 2000

Partial sums of A023416. - Reinhard Zumkeller, Jul 15 2011

The graph of this sequence is a version of the Takagi curve: see Lagarias (2012), Section 9, especially Theorem 9.1. - N. J. A. Sloane, Mar 12 2016

T. D. Noe and Hieronymus Fischer, Table of n, a(n) for n = 0..10000 (terms up to n=1023 by T. D. Noe)
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Jeffrey C. Lagarias, The Takagi function and its properties, arXiv:1112.4205 [math.CA], 2011-2012.
Jeffrey C. Lagarias, The Takagi function and its properties, In Functions in number theory and their probabilistic aspects, 153--189, RIMS Kôkyûroku Bessatsu, B34, Res. Inst. Math. Sci. (RIMS), Kyoto, 2012. MR3014845.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences related to binary expansion of n

The basic sequences concerning the binary expansion of n are A000120, A000788, A000069, A001969, A023416, A059015, A070939, A083652.

Cf. A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564 (for base 10).

a059015 n = a059015_list !! n
a059015_list = scanl1 (+) $ map a023416 [0..]
-- Reinhard Zumkeller, Jul 15 2011

a:= proc(n) option remember; `if`(n=0, 1, a(n-1)+add(1-i, i=Bits[Split](n))) end:
seq(a(n), n=0..65);  # Alois P. Heinz, Nov 11 2024

Accumulate[ Table[ Count[ IntegerDigits[n, 2], 0], {n, 0, 65}]] (* Jean-François Alcover, Oct 03 2012 *)
Accumulate[DigitCount[Range[0,70],2,0]] (* Harvey P. Dale, Jun 24 2017 *)

v=vector(100,i,1);for(i=1,#v-1,v[i+1] = v[i] + #binary(i) - hammingweight(i)); v \\ Charles R Greathouse IV, Nov 20 2012

a(n)=if(n, my(m=logint(n,2)); 2 + (m+1)*(n+1) - 2^(m+1) + sum(j=1,m+1, my(t=floor(n/2^j + 1/2)); (n>>j)*(2*n + 2 - (1 + (n>>j))<Charles R Greathouse IV, Dec 14 2015

def A059015(n): return 2+(n+1)*(m:=(n+1).bit_length())-(1<Chai Wah Wu, Mar 01 2023

def A059015(n): return 2+(n+1)*((t:=(n+1).bit_length())-n.bit_count())-(1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1))>>1) # Chai Wah Wu, Nov 11 2024

a(n) = b(n)+1, with b(2n) = b(n)+b(n-1)+n, b(2n+1) = 2b(n)+n. - Ralf Stephan, Sep 13 2003
From Hieronymus Fischer, Jun 10 2012: (Start)
With m = floor(log_2(n)):
a(n) = 2 + (m+1)*(n+1) - 2^(m+1) + (1/2)*Sum_{j=1..m+1} (floor(n/2^j)*(2*n + 2 - (1 + floor(n/2^j))*2^j) - floor(n/2^j + 1/2)*(2*n + 2 - floor(n/2^j + 1/2)*2^j)).
a(n) = A083652(n) - (n+1)*A000120(n) + 2^(m-1) - (1/4) + (1/2)*sum_{j=1..m+1} (floor(n/2^j + 1/2)^2 - (floor(n/2^j) + 1/2)^2)*2^j.
a(2^m-1) = 2 + (m-2)*2^(m-1)
(this is the total number of zero digits occurring in all the numbers with <= m places).
G.f.: 1/(1 - x) + (1/(1 - x)^2)*Sum_{j>=0} x^(2*2^j)/(1 + x^(2^j)); corrected by Ilya Gutkovskiy, Mar 28 2018
General formulas for the number of digits <= d in the base p representations of all integers from 0 to n, where 0 <= d < p.
With m = floor(log_p(n)):
a(n) = 1 + (m+1)*(n+1) - (p^(m+1)-1)/(p-1) + (1/2)*sum_{j=1..m+1} (floor(n/p^j)*(2n + 2 - (1 + floor(n/p^j))*p^j) - floor(n/p^j + (p-d-1)/p)*(2n + 2 + ((p-2*d-2)/p - floor(n/p^j + (p-d-1)/p))*p^j)).
a(n) = H(n,p) - (n+1)*F(n,p,d+1) + (1/2)*sum_{j=1..m+1} ((floor(n/p^j + (p-d-1)/p)^2 - floor(n/p^j)^2)*p^j - (((p - 2*d-2)/p)*floor(n/p^j + (p-d-1)/p) + floor(n/p^j))*p^j), where H(n,p) = sum of number of digits in the base p representations of 0 to n and F(n,p,d) = number of digits >=d in the base p representation of n.
a(p^m-1) = 1 + (d+1)*m*p^(m-1) - (p^m-1)/(p-1).
(this is the total number of digits <= d occurring in all the numbers with <= m places in base p representation).
G.f.: 1/(1-x) + (1/(1-x)^2)*Sum_{j>=0} ((1-x^(d*p^j))*x^p^j + (1-x^p^j)*x^p^(j+1)/(1-x^p^(j+1))). (End)

A037861 (Number of 0's) - (number of 1's) in the base-2 representation of n.

Original entry on oeis.org

1, -1, 0, -2, 1, -1, -1, -3, 2, 0, 0, -2, 0, -2, -2, -4, 3, 1, 1, -1, 1, -1, -1, -3, 1, -1, -1, -3, -1, -3, -3, -5, 4, 2, 2, 0, 2, 0, 0, -2, 2, 0, 0, -2, 0, -2, -2, -4, 2, 0, 0, -2, 0, -2, -2, -4, 0, -2, -2, -4, -2, -4, -4, -6, 5, 3, 3, 1, 3, 1, 1, -1, 3

Offset: 0

Clark Kimberling

-Sum_{n>=1} a(n)/((2*n)*(2*n+1)) = the "alternating Euler constant" log(4/Pi) = 0.24156... - (see A094640 and Sondow 2005, 2010).

a(A072600(n)) < 0; a(A072601(n)) <= 0; a(A031443(n)) = 0; a(A072602(n)) >= 0; a(A072603(n)) > 0; a(A031444(n)) = 1; a(A031448(n)) = -1; abs(a(A089648(n))) <= 1. - Reinhard Zumkeller, Feb 07 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 112 (2005), 61-65.
Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005; Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions (ps file).
Ralf Stephan, Table of generating functions (pdf file).
Index entries for sequences related to binary expansion of n

Cf. A031443 for n when a(n)=0, A053738 for n when a(n) odd, A053754 for n when a(n) even, A030300 for a(n+1) mod 2.
Cf. A066879, A094640, A110625, A145037.
Cf. A072600, A072601, A072602, A072603, A031444, A031448, A089648.
See A268289 for a recurrence based on this sequence.

a037861 n = a023416 n - a000120 n  -- Reinhard Zumkeller, Aug 01 2013

A037861:= proc(n) local L;
     L:= convert(n,base,2);
     numboccur(0,L) - numboccur(1,L)
end proc:
map(A037861, [$0..100]); # Robert Israel, Mar 08 2016

Table[Count[ IntegerDigits[n, 2], 0] - Count[IntegerDigits[n, 2], 1], {n, 0, 75}]

a(n) = if (n==0, 1, 1 + logint(n, 2) - 2*hammingweight(n)); \\ Michel Marcus, May 15 2020 and Jun 16 2020

def A037861(n):
    return 2*format(n,'b').count('0')-len(format(n,'b')) # Chai Wah Wu, Mar 07 2016

From Henry Bottomley, Oct 27 2000: (Start)
a(n) = A023416(n) - A000120(n) = A029837(n) - 2*A000120(n) = 2*A023416(n) - A029837(n).
a(2*n) = a(n) + 1; a(2*n + 1) = a(2*n) - 2 = a(n) - 1. (End)
G.f. satisfies A(x) = (1 + x)*A(x^2) - x*(2 + x)/(1 + x). - Franklin T. Adams-Watters, Dec 26 2006
a(n) = b(n) for n > 0 with b(0) = 0 and b(n) = b(floor(n/2)) + (-1)^(n mod 2). - Reinhard Zumkeller, Dec 31 2007
G.f.: 1 + (1/(1 - x))*Sum_{k>=0} x^(2^k)*(x^(2^k) - 1)/(1 + x^(2^k)). - Ilya Gutkovskiy, Apr 07 2018

A020522 a(n) = 4^n - 2^n.

Original entry on oeis.org

0, 2, 12, 56, 240, 992, 4032, 16256, 65280, 261632, 1047552, 4192256, 16773120, 67100672, 268419072, 1073709056, 4294901760, 17179738112, 68719214592, 274877382656, 1099510579200, 4398044413952, 17592181850112, 70368735789056, 281474959933440

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of walks of length 2*n+2 between any two diametrically opposite vertices of the cycle graph C_8. - Herbert Kociemba, Jul 02 2004
If we consider a(4*k+2), then 2^4 == 3^4 == 3 (mod 13); 2^(4*k+2) + 3^(4*k+2) == 3^k*(4+9) == 3*0 == 0 (mod 13). So a(4*k+2) can never be prime. - Jose Brox, Dec 27 2005
If k is odd, then a(n*k) is divisible by a(n), since: a(n*k) = (2^n)^k + (3^n)^k = (2^n + 3^n)*((2^n)^(k-1) - (2^n)^(k-2) (3^n) + - ... + (3^n)^(k-1)). So the only possible primes in the sequence are a(0) and a(2^n) for n>=1. I've checked that a(2^n) is composite for 3 <= n <= 15. As with Fermat primes, a probabilistic argument suggests that there are only finitely many primes in the sequence. - Dean Hickerson, Dec 27 2005
Let x,y,z be elements from some power set P(n), i.e., the power set of a set of n elements. Define a function f(x,y,z) in the following manner: f(x,y,z) = 1 if x is a subset of y and y is a subset of z and x does not equal z; f(x,y,z) = 0 if x is not a subset of y or y is not a subset of z or x equals z. Now sum f(x,y,z) for all x,y,z of P(n). This gives a(n). - Ross La Haye, Dec 26 2005
Number of monic (irreducible) polynomials of degree 1 over GF(2^n). - Max Alekseyev, Jan 13 2006
Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then a(n) = the number of (x,y) of B for which x does not equal y. - Ross La Haye, Jan 02 2008
For n>1: central terms of the triangle in A173787. - Reinhard Zumkeller, Feb 28 2010
Pronic numbers of the form: (2^n - 1)*2^n, which is the n-th Mersenne number times 2^n, see A000225 and A002378. - Fred Daniel Kline, Nov 30 2013
Indices where records of A037870 occur. - Philippe Beaudoin, Sep 03 2014
Half the total edge length for a minimum linear arrangement of a hypercube of dimension n. (See Harper's paper below for proof). - Eitan Frachtenberg, Apr 07 2017
Number of pairs in GF(2)^{n+1} whose dot product is 1. - Christopher Purcell, Dec 11 2021

			n=5: a(5) = 4^5 - 2^5 = 1024 - 32 = 992 -> '1111100000'.

Vincenzo Librandi, Table of n, a(n) for n = 0..170
M. Archibald, A. Blecher, A. Knopfmacher, and M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
Tom Copeland, The Kervaire-Milnor formula
John Elias, Illustration of initial terms: Twin 2^n hexagonal numbers
L. H. Harper, Optimal Assignment of Numbers to Vertices, J. SIAM 12(1), p. 131--135, March 1964; alternative link.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
The Sixtieth William Lowell Putnam Mathematical Competition, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Ratio of successive terms of A028365.
Cf. A000225, A060867, A161168, A006516, A059153, A065442.
Cf. A000079, A265736, A020988, A047849.
Cf. A000079, A000302.

a020522 = (* 2) . a006516  -- Reinhard Zumkeller, Dec 15 2015

[4^n - 2^n: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

A020522:=n->4^n-2^n; seq(A020522(n), n=0..50); # Wesley Ivan Hurt, Nov 29 2013

Table[4^n - 2^n, {n, 40}] (* or *) LinearRecurrence[{6, -8}, {0, 2}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)

a(n)=4^n-2^n \\ Charles R Greathouse IV, Jan 30 2012

def A020522(n): return (1<Chai Wah Wu, Mar 10 2025

[4^n - 2^n for n in range(0,23)] # Zerinvary Lajos, Jun 05 2009

From Herbert Kociemba, Jul 02 2004: (Start)
G.f.: 2*x/((-1 + 2*x)*(-1 + 4*x)).
a(n) = 6*a(n-1) - 8*a(n-2). (End)
E.g.f.: exp(4*x) - exp(2*x). - Mohammad K. Azarian, Jan 14 2009
From Reinhard Zumkeller, Feb 07 2006, Jaroslav Krizek, Aug 02 2009: (Start)
a(n) = A099393(n)-A000225(n+1) = A083420(n)-A099393(n).
In binary representation, n>0: n 1's followed by n 0's (A138147(n)).
A000120(a(n)) = n.
A023416(a(n)) = n.
A070939(a(n)) = 2*n.
2*a(n)+1 = A030101(A099393(n)). (End)
a(n) = A085812(n) - A001700(n). - John Molokach, Sep 28 2013
a(n) = 2*A006516(n) = A000079(n)*A000225(n) = A265736(A000225(n)). - Reinhard Zumkeller, Dec 15 2015
a(n) = (4^(n/2) - 4^(n/4))*(4^(n/2) + 4^(n/4)). - Bruno Berselli, Apr 09 2018
Sum_{n>0} 1/a(n) = E - 1, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022
a(n) = A000302(n) - A000079(n). - John Reimer Morales, Aug 04 2025

A055640 Number of nonzero digits in decimal expansion of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2

Offset: 0

Henry Bottomley, Jun 06 2000

Comment from Antti Karttunen, Sep 05 2004: (Start)
Also number of characters needed to write the number n in classical Greek alphabetic system, up to n=999. The Greek alphabetic system assigned values to the letters as follows:
alpha = 1, beta = 2, gamma = 3, delta = 4, epsilon = 5, digamma = 6, zeta = 7, eta = 8, theta = 9, iota = 10, kappa = 20, lambda = 30, mu = 40, nu = 50, xi = 60, omicron = 70, pi = 80, koppa = 90, rho = 100, sigma = 200, tau = 300, upsilon = 400, phi = 500, chi = 600, psi = 700, omega = 800, sampi = 900. (End)
For partial sums see A102685. - Hieronymus Fischer, Jun 06 2012

			129 is written as rho kappa theta in the old Greek system.

L. Threatte, The Greek Alphabet, in The World's Writing Systems, edited by Peter T. Daniels and William Bright, Oxford Univ. Press, 1996, p. 278.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000
Unicode Consortium, Unicode Home Page. (Follow the link "Display Problems?" to find an appropriate information/font file to show the Greek characters correctly. Or look at the HTML source to see their names.)

Cf. A102669-A102685, A004719, A011540, A052382, A061745, A054899, A055641, A055642, A122840, A160093, A160094, A193238, A196563, A195564, A000120, A000788, A023416, A059015 (for base 2).

Differs from A098378 for the first time at position n=200 with a(200)=1, as only one nonzero Arabic digit (and only one Greek letter) is needed for two hundred, while A098378(200)=2 as two characters are needed in the Ethiopic system.

a055640 n = length $ filter (/= '0') $ show n
-- Reinhard Zumkeller, May 02 2011

Table[Count[IntegerDigits[n],?(#>0&)],{n,0,120}] (* _Harvey P. Dale, Mar 11 2012 *)
Total[Most[DigitCount[#]]]&/@Range[0,120] (* Harvey P. Dale, Mar 19 2021 *)

a(n)=my(v=digits(n));sum(i=1,#v,!!v[i]) \\ Charles R Greathouse IV, Aug 05 2012

From Hieronymus Fischer, Jun 06 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j+0.9) - floor(n/10^j)), where m = floor(log_10(n)).
a(n) = m + 1 - A055641(n).
G.f.: (1/(1-x))*Sum_{j>=0} (x^10^j - x^(10*10^j))/(1-x^10^(j+1)). (End)
a(n) = A055642(n) - A055641(n).

cp's OEIS Frontend

A031443 Digitally balanced numbers: positive numbers that in base 2 have the same number of 0's as 1's.

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A080791 Number of nonleading 0's in binary expansion of n.

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A000788 Total number of 1's in binary expansions of 0, ..., n.

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A055641 Number of zero digits in n.

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A003754 Numbers with no adjacent 0's in binary expansion.

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