A079944 -id:A079944 - cp's OEIS frontend

A069010 Number of runs of 1's in the binary representation of n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 2, 3, 3, 3, 2, 3

Offset: 0

Henry Bottomley, Apr 02 2002

a(n) is also the number of distinct parts in the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017

Positions of first occurrences of k are A002450(k). - John Keith, Aug 30 2021

			a(11) = 2 since 11 is 1011 in binary with two runs of 1's.
a(12) = 1 since 12 is 1100 in binary with one run of 1's.

Tilman Piesk (terms 0..9999) & Antti Karttunen, Table of n, a(n) for n = 0..16384
Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 148.
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A000035, A000120, A005811, A014081, A033264, A037800.
Cf. A268411 (parity of the terms), A268412 (positions of even terms), A268415 (of odd terms).
Cf. A002450 (positions of record highs).
Cf. also A227349, A246588.

f:= proc(n) option remember; if n::even then procname(n/2)
elif n mod 4 = 1 then 1 + procname((n-1)/2) else  procname((n-1)/2) fi end proc:
f(0):= 0:
map(f, [$0..1000]); # Robert Israel, Sep 06 2015

Count[Split@ IntegerDigits[#, 2], n_ /; First@ n == 1] & /@ Range[0, 120] (* Michael De Vlieger, Sep 05 2015 *)

a(n) = (1 + (hammingweight(bitxor(n, n>>1)))) >> 1;  \\ Gheorghe Coserea, Sep 05 2015

def A069010(n):
    return sum(1 for d in bin(n)[2:].split('0') if len(d)) # Chai Wah Wu, Nov 04 2016

(define (A069010 n) (/ (+ (A005811 n) (A000035 n)) 2)) ;; Antti Karttunen, Feb 05 2016

a(n) = ceiling(A005811(n)/2) = A005811(n) - A033264(n). If 2^k <= n < 3*2^(k-1) then a(n) = a(n-2^k)+1; if 3*2^(k-1) <= n < 2^(k+1) then a(n) = a(n-2^k).
a(2n) = a(n), a(2n+1) = a(n) + [n is even]. - Ralf Stephan, Aug 20 2003
G.f.: (1/(1-x)) * Sum_{k>=0} (t/(1+t))/(1+t^2), where t=x^2^k. - Ralf Stephan, Sep 07 2003
a(n) = A000120(n) - A014081(n) = A037800(n) + 1, n>0. - Ralf Stephan, Sep 10 2003

A079000 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is odd".

Original entry on oeis.org

1, 4, 6, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 20, 21, 23, 25, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 97

Offset: 1

Matthew Vandermast, Feb 01 2003

a(a(n)) = 2n + 3 for n>1.

			a(2) cannot be 2 because 2 is even; it cannot be 3 because that would require 2 to be a member of the sequence. Hence a(2)=4 and the next odd member of the sequence is the fourth member.

Hsien-Kuei Hwang, S Janson, TH 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; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also 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
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
N. J. A. Sloane, Seven Staggering Sequences.
Index entries for sequences of the a(a(n)) = 2n family

Cf. A079250-A079259, A079313, A079325, A064437, A003605, A079352, A079358.
Cf. also A080596, A080731, A080752, A121053, A080032.
Partial sums give A080566. Differences give A079948.

Digits := 50; A079000 := proc(n) local k,j; if n<=2 then n^2; else k := floor(evalf(log( (n+3)/6 )/log(2)) ); j := n-(9*2^k-3); 12*2^k-3+3*j/2 +abs(j)/2; fi; end;
A002264 := n->floor(n/3): A079944 := n->floor(log[2](4*(n+2)/3))-floor(log[2](n+2)): A000523 := n->floor(log[2](n)): f := n->A079944(A002264(n-4)): g := n->A000523(A002264(n+2)/2): A079000 := proc(n) if n>3 then RETURN(simplify(3*n+3-3*2^g(n)+(-1)^f(n)*(9*2^g(n)-n-3))/2) else if n>0 then RETURN([1,4,6][n]) else RETURN(0) fi fi: end;

a[1] = 1; a[n_] := (k = Floor[Log[2, (n+3)/6]]; j = n-(9*2^k - 3); 12*2^k-3 + 3*j/2 + Abs[j]/2); Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 21 2012, after Maple *)

a(1) = 1, a(2) = 4, then a(9*2^k-3+j) = 12*2^k-3+3*j/2+|j|/2 for k>=0, -3*2^k <= j <= 3*2^k. Also a(3n) = 3*b(n/3), a(3n+1) = 2*b(n)+b(n+1), a(3n+2) = b(n)+2*b(n+1) for n>=2, where b = A079905. - N. J. A. Sloane and Benoit Cloitre, Feb 20 2003
a(n+1) - 2*a(n) + a(n-1) = 1 for n = 9*2^k - 3, k>=0, = -1 for n = 2 and 3*2^k-3, k>=1 and = 0 otherwise.
a(n) = (3*n + 3 - 3*2^g(n) + (-1)^f(n)*(9*2^g(n) - n - 3))/2 for n>3, f(n) = A079944(A002264(n-4)) and g(n) = A000523(A002264(n+2)/2). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003
Also a(n) = n + 3*2^A000523(A002264(n+2)/2)*(1 - 3*A080584(n-4)) + A080584(n-4)*(n+3) for n>3, where A080584(n)=A079944(A002264(n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 24 2003

A035263 Trajectory of 1 under the morphism 0 -> 11, 1 -> 10; parity of 2-adic valuation of 2n: a(n) = A000035(A001511(n)).

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1

Offset: 1

Karamanos Konstantinos, N. J. A. Sloane

First Feigenbaum symbolic (or period-doubling) sequence, corresponding to the accumulation point of the 2^{k} cycles through successive bifurcations.
To construct the sequence: start with 1 and concatenate: 1,1, then change the last term (1->0; 0->1) gives: 1,0. Concatenate those 2 terms: 1,0,1,0, change the last term: 1,0,1,1. Concatenate those 4 terms: 1,0,1,1,1,0,1,1 change the last term: 1,0,1,1,1,0,1,0, etc. - Benoit Cloitre, Dec 17 2002
Let T denote the present sequence. Here is another way to construct T. Start with the sequence S = 1,0,1,,1,0,1,,1,0,1,,1,0,1,,... and fill in the successive holes with the successive terms of the sequence T (from paper by Allouche et al.). - Emeric Deutsch, Jan 08 2003 [Note that if we fill in the holes with the terms of S itself, we get A141260. - N. J. A. Sloane, Jan 14 2009]
From N. J. A. Sloane, Feb 27 2009: (Start)
In more detail: define S to be 1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1,0,1___...
If we fill the holes with S we get A141260:
1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0,
........1.........0.........1.........1.........0.......1.........1.........0...
- the result is
1..0..1.1.1..0..1.0.1..0..1.1.1..0..1.1.1..0..1.0.1.... = A141260.
But instead, if we define T recursively by filling the holes in S with the terms of T itself, we get A035263:
1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0,
........1.........0.........1.........1.........1.......0.........1.........0...
- the result is
1..0..1.1.1..0..1.0.1..0..1.1.1..0..1.1.1..0..1.1.1.0.1.0.1..0..1.1.1..0..1.0.1.. = A035263. (End)
Characteristic function of A003159, i.e., A035263(n)=1 if n is in A003159 and A035263(n)=0 otherwise (from paper by Allouche et al.). - Emeric Deutsch, Jan 15 2003
This is the sequence of R (=1), L (=0) moves in the Towers of Hanoi puzzle: R, L, R, R, R, L, R, L, R, L, R, R, R, ... - Gary W. Adamson, Sep 21 2003
Manfred Schroeder, p. 279 states, "... the kneading sequences for unimodal maps in the binary notation, 0, 1, 0, 1, 1, 1, 0, 1..., are obtained from the Morse-Thue sequence by taking sums mod 2 of adjacent elements." On p. 278, in the chapter "Self-Similarity in the Logistic Parabola", he writes, "Is there a closer connection between the Morse-Thue sequence and the symbolic dynamics of the superstable orbits? There is indeed. To see this, let us replace R by 1 and C and L by 0." - Gary W. Adamson, Sep 21 2003
Partial sums modulo 2 of the sequence 1, a(1), a(1), a(2), a(2), a(3), a(3), a(4), a(4), a(5), a(5), a(6), a(6), ... . - Philippe Deléham, Jan 02 2004
Parity of A007913, A065882 and A065883. - Philippe Deléham, Mar 28 2004
The length of n-th run of 1's in this sequence is A080426(n). - Philippe Deléham, Apr 19 2004
Also parity of A005043, A005773, A026378, A104455, A117641. - Philippe Deléham, Apr 28 2007
Equals parity of the Towers of Hanoi, or ruler sequence (A001511), where the Towers of Hanoi sequence (1, 2, 1, 3, 1, 2, 1, 4, ...) denotes the disc moved, labeled (1, 2, 3, ...) starting from the top; and the parity of (1, 2, 1, 3, ...) denotes the direction of the move, CW or CCW. The frequency of CW moves converges to 2/3. - Gary W. Adamson, May 11 2007
A conjectured identity relating to the partition sequence, A000041: p(x) = A(x) * A(x^2) when A(x) = the Euler transform of A035263 = polcoeff A174065: (1 + x + x^2 + 2x^3 + 3x^4 + 4x^5 + ...). - Gary W. Adamson, Mar 21 2010
a(n) is 1 if the number of trailing zeros in the binary representation of n is even. - Ralf Stephan, Aug 22 2013
From Gary W. Adamson, Mar 25 2015: (Start)
A conjectured identity relating to the partition sequence, A000041 as polcoeff p(x); A003159, and its characteristic function A035263: (1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, ...); and A036554 indicating n-th terms with zeros in A035263: (2, 6, 8, 10, 14, 18, 22, ...).
The conjecture states that p(x) = A(x) = A(x^2) when A(x) = polcoeffA174065 = the Euler transform of A035263 = 1/(1-x)*(1-x^3)*(1-x^4)*(1-x^5)*... = (1 + x + x^2 + 2x^3 + 3x^4 + 4x^5 + ...) and the aerated variant = the Euler transform of the complement of A035263: 1/(1-x^2)*(1-x^6)*(1-x^8)*... = (1 + x^2 + x^4 + 2x^6 + 3x^8 + 4x^10 + ...).
(End)
The conjecture above was proved by Jean-Paul Allouche on Dec 21 2013.
Regarded as a column vector, this sequence is the product of A047999 (Sierpinski's gasket) regarded as an infinite lower triangular matrix and A036497 (the Fredholm-Rueppel sequence) where the 1's have alternating signs, 1, -1, 0, 1, 0, 0, 0, -1, .... - Gary W. Adamson, Jun 02 2021
The numbers of 1's through n (A050292) can be determined by starting with the binary (say for 19 = 1 0 0 1 1) and writing: next term is twice current term if 0, otherwise twice plus 1. The result is 1, 2, 4, 9, 19. Take the difference row, = 1, 1, 2, 5, 10; and add the odd-indexed terms from the right: 5, 4, 3, 2, 1 = 10 + 2 + 1 = 13. The algorithm is the basis for determining the disc configurations in the tower of Hanoi game, as shown in the Jul 24 2021 comment of A060572. - Gary W. Adamson, Jul 28 2021

Karamanos, Kostas. "From symbolic dynamics to a digital approach." International Journal of Bifurcation and Chaos 11.06 (2001): 1683-1694. (Full version. See p. 1685)
Karamanos, K. (2000). From symbolic dynamics to a digital approach: chaos and transcendence. In Michel Planat (Ed.), Noise, Oscillators and Algebraic Randomness (Lecture Notes in Physics, pp. 357-371). Springer, Berlin, Heidelberg. (Short version. See p. 359)
Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W. H. Freeman, 1991
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 892, column 2, Note on p. 84, part (a).

Antti Karttunen, Table of n, a(n) for n = 1..65536 (first 10000 terms from T. D. Noe)
Ricardo Gómez Aíza, Symbolic dynamical scales: modes, orbitals, and transversals, arXiv:2009.02669 [math.DS], 2020.
Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, CNRS France, 2024. See p. 6.
Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, Preprint, 2024 [Local copy, with permission]
Jean-Paul Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., 139 (1995), 455-461.
Jean-Paul Allouche and Michel Mendès France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11.
Jean-Paul Allouche and Michel Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. [Local copy]
Jean-Paul Allouche and Jeffrey Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
Joerg Arndt, Matters Computational (The Fxtbook), pp.9-10, pp.734-738
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
Benoit Cloitre, Fractal walk on the 130000 first terms (step of unit length moving with angle Pi/3 if 0 and -Pi/3 if 1) starting at (0,0)
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
B. Derrida, A. Gervois and Y. Pomeau, Iteration of endomorphisms on the real axis and representation of number, Ann. Inst. Henri Poincaré, Section A: Physique Théorique, Vol. XXIX no. 3, 305-356 (1978).
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004.
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović, and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 79. Book's website
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.
Konstantinos Karamanos, Symbolic Dynamics to a Digital Approach, Int. J. of Bifurcation and Chaos, 11(6), 1683-1694 (2001).
Konstantinos Karamanos and Grégoire Nicolis, Symbolic Dynamics and Entropy Analysis of Feigenbaum Limit Sets, Chaos, Solitons and Fractals 10(7), 1135 - 1150 (1999).
D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions
Joseph Meleshko, Pascal Ochem, Jeffrey Shallit, and Sonja Linghui Shan, Pseudoperiodic Words and a Question of Shevelev, arXiv:2207.10171 [math.CO], 2022.
Nicholas C. Metropolis, Myron L. Stein and Paul R. Stein, On finite limit sets for transformations on the unit interval, J. Combin. Theory, A 15 (1973), 25-44.
Jeffrey Shallit and Anatoly Zavyalov, Transduction of Automatic Sequences and Applications, arXiv:2303.15203 [cs.FL], 2023, see p. 5.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Eric Weisstein's World of Mathematics, Binary carry sequence.
Eric Weisstein's World of Mathematics, Double-free set.
Index entries for characteristic functions
Index entries for sequences related to binary expansion of n
Index entries for sequences that are fixed points of mappings

Parity of A001511. Anti-parity of A007814.
Absolute values of first differences of A010060. Apart from signs, same as A029883. Essentially the same as A056832.
Swapping 0 and 1 gives A096268.
Cf. A033485, A050292 (partial sums), A089608, A088172, A019300, A039982, A073675, A121701, A141260, A000041, A174065, A220466, A154269 (Mobius transform).
Limit of A317957(n) for large n.
Cf. A060572

import Data.Bits (xor)
a035263 n = a035263_list !! (n-1)
a035263_list = zipWith xor a010060_list $ tail a010060_list
-- Reinhard Zumkeller, Mar 01 2012

nmax:=105: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (p+1) mod 2 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 07 2013
A035263 := n -> 1 - padic[ordp](n, 2) mod 2:
seq(A035263(n), n=1..105); # Peter Luschny, Oct 02 2018

a[n_] := a[n] = If[ EvenQ[n], 1 - a[n/2], 1]; Table[ a[n], {n, 1, 105}] (* Or *)
Rest[ CoefficientList[ Series[ Sum[ x^(2^k)/(1 + (-1)^k*x^(2^k)), {k, 0, 20}], {x, 0, 105}], x]]
f[1] := True; f[x_] := Xor[f[x - 1], f[Floor[x/2]]]; a[x_] := Boole[f[x]] (* Ben Branman, Oct 04 2010 *)
a[n_] := If[n == 0, 0, 1 - Mod[ IntegerExponent[n, 2], 2]]; (* Jean-François Alcover, Jul 19 2013, after Michael Somos *)
Nest[ Flatten[# /. {0 -> {1, 1}, 1 -> {1, 0}}] &, {0}, 7] (* Robert G. Wilson v, Jul 23 2014 *)
SubstitutionSystem[{0->{1,1},1->{1,0}},1,{7}][[1]] (* Harvey P. Dale, Jun 06 2022 *)

{a(n) = if( n==0, 0, 1 - valuation(n, 2)%2)}; /* Michael Somos, Sep 04 2006 */

{a(n) = if( n==0, 0, n = abs(n); subst( Pol(binary(n)) - Pol(binary(n-1)), x, 1)%2)}; /* Michael Somos, Sep 04 2006 */

{a(n) = if( n==0, 0, n = abs(n); direuler(p=2, n, 1 / (1 - X^((p<3) + 1)))[n])}; /* Michael Somos, Sep 04 2006 */

def A035263(n): return (n&-n).bit_length()&1 # Chai Wah Wu, Jan 09 2023

(define (A035263 n) (let loop ((n n) (i 1)) (cond ((odd? n) (modulo i 2)) (else (loop (/ n 2) (+ 1 i)))))) ;; (Use mod instead of modulo in R6RS) Antti Karttunen, Sep 11 2017

Absolute values of first differences (A029883) of Thue-Morse sequence (A001285 or A010060). Self-similar under 10->1 and 11->0.
Series expansion: (1/x) * Sum_{i>=0} (-1)^(i+1)*x^(2^i)/(x^(2^i)-1). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 17 2003
a(n) = Sum_{k>=0} (-1)^k*(floor((n+1)/2^k)-floor(n/2^k)). - Benoit Cloitre, Jun 03 2003
Another g.f.: Sum_{k>=0} x^(2^k)/(1+(-1)^k*x^(2^k)). - Ralf Stephan, Jun 13 2003
a(2*n) = 1-a(n), a(2*n+1) = 1. - Ralf Stephan, Jun 13 2003
a(n) = parity of A033485(n). - Philippe Deléham, Aug 13 2003
Equals A088172 mod 2, where A088172 = 1, 2, 3, 7, 13, 26, 53, 106, 211, 422, 845, ... (first differences of A019300). - Gary W. Adamson, Sep 21 2003
a(n) = a(n-1) - (-1)^n*a(floor(n/2)). - Benoit Cloitre, Dec 02 2003
a(1) = 1 and a(n) = abs(a(n-1) - a(floor(n/2))). - Benoit Cloitre, Dec 02 2003
a(n) = 1 - A096268(n+1); A050292 gives partial sums. - Reinhard Zumkeller, Aug 16 2006
Multiplicative with a(2^k) = 1 - (k mod 2), a(p^k) = 1, p > 2. Dirichlet g.f.: Product_{n = 4 or an odd prime} (1/(1-1/n^s)). - Christian G. Bower, May 18 2005
a(-n) = a(n). a(0)=0. - Michael Somos, Sep 04 2006
Dirichlet g.f.: zeta(s)*2^s/(2^s+1). - Ralf Stephan, Jun 17 2007
a(n+1) = a(n) XOR a(ceiling(n/2)), a(1) = 1. - Reinhard Zumkeller, Jun 11 2009
Let D(x) be the generating function, then D(x) + D(x^2) == x/(1-x). - Joerg Arndt, May 11 2010
a(n) = A010060(n) XOR A010060(n+1); a(A079523(n)) = 0; a(A121539(n)) = 1. - Reinhard Zumkeller, Mar 01 2012
a((2*n-1)*2^p) = (p+1) mod 2, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 07 2013
a(n) = A000035(A001511(n)). - Omar E. Pol, Oct 29 2013
a(n) = 2-A056832(n) = (5-A089608(n))/4. - Antti Karttunen, Sep 11 2017, after Benoit Cloitre
For n >= 0, a(n+1) = M(2n) mod 2 where M(n) is the Motzkin number A001006 (see Deutsch and Sagan 2006 link). - David Callan, Oct 02 2018
a(n) = A038712(n) mod 3. - Kevin Ryde, Jul 11 2019
Given any n in the form (k * 2^m, k odd), extract k and m. Categorize the results into two outcomes of (k, m, even or odd). If (k, m) is (odd, even) substitute 1. If (odd, odd), denote the result 0.  Example: 5 = (5 * 2^0), (odd, even, = 1). (6 = 3 * 2^1), (odd, odd, = 0). - Gary W. Adamson, Jun 23 2021

Alternative description added to the name by Antti Karttunen, Sep 11 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.
Julien Leroy, Michel Rigo, and Manon Stipulanti, Behavior of Digital Sequences Through Exotic Numeration Systems, Electronic Journal of Combinatorics 24(1) (2017), #P1.44.
Raoul Nakhmanson-Kulish, Graphic representation of K(n)=a(n)/(n log2(n)/2) from n=63 to 131071
D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc., 21 (1969), pp. 719-721. [From the bibliography of Stolarsky, 1977]
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
S. C. Tang, An improvement and generalization of Bellman-Shapiro's theorem on a problem in additive number theory, Proc. Amer. Math. Soc., 14 (1963), pp. 199-204. [From the bibliography of Stolarsky, 1977]
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

A025480 a(2n) = n, a(2n+1) = a(n).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 0, 4, 2, 5, 1, 6, 3, 7, 0, 8, 4, 9, 2, 10, 5, 11, 1, 12, 6, 13, 3, 14, 7, 15, 0, 16, 8, 17, 4, 18, 9, 19, 2, 20, 10, 21, 5, 22, 11, 23, 1, 24, 12, 25, 6, 26, 13, 27, 3, 28, 14, 29, 7, 30, 15, 31, 0, 32, 16, 33, 8, 34, 17, 35, 4, 36, 18, 37, 9, 38, 19, 39, 2, 40, 20, 41, 10

Offset: 0

David W. Wilson

These are the Grundy values or nim-values for heaps of n beans in the game where you're allowed to take up to half of the beans in a heap. - R. K. Guy, Mar 30 2006. See Levine 2004/2006 for more about this. - N. J. A. Sloane, Aug 14 2016
When n > 0 is written as (2k+1)*2^j then k = a(n-1) and j = A007814(n), so: when n is written as (2k+1)*2^j-1 then k = a(n) and j = A007814(n+1), when n > 1 is written as (2k+1)*2^j+1 then k = a(n-2) and j = A007814(n-1). - Henry Bottomley, Mar 02 2000 [sequence id corrected by Peter Munn, Jun 22 2022]
According to the comment from Deuard Worthen (see Example section), this may be regarded as a triangle where row r=1,2,3,... has length 2^(r-1) and values T(r,2k-1)=T(r-1,k), T(r,2k)=2^(r-1)+k-1; i.e., previous row gives 1st, 3rd, 5th, ... term and 2nd, 4th, ... terms are numbers 2^(r-1),...,2^r-1 (i.e., those following the last one from the previous row). - M. F. Hasler, May 03 2008
Let StB be a Stern-Brocot tree hanging between (pseudo)fractions Left and Right, then StB(1) = mediant(Left,Right) and for n>1: StB(n) = if a(n-1)<>0 and a(n)<>0 then mediant(StB(a(n-1)),StB(a(n))) else if a(n)=0 then mediant(StB(a(n-1)),Right) else mediant(Left,StB(a(n-1))), where mediant(q1,q2) = ((numerator(q1)+numerator(q2)) / (denominator(q1)+denominator(q2))). - Reinhard Zumkeller, Dec 22 2008
This sequence is the unique fixed point of the function (a(0), a(1), a(2), ...) |--> (0, a(0), 1, a(1), 2, a(2), ...) which interleaves the nonnegative integers between the elements of a sequence. - Cale Gibbard (cgibbard(AT)gmail.com), Nov 18 2009
Also the number of remaining survivors in a Josephus problem after the person originally first in line has been eliminated (see A225381). - Marcus Hedbring, May 18 2013
A fractal sequence - see Levine 2004/2006. - N. J. A. Sloane, Aug 14 2016
From David James Sycamore, Apr 29 2020: (Start)
One of a family of fractal sequences, S_k; defined as follows for k >= 2: a(k*n) = n, a(k*n+r) = a((k-1)*n + (r-1)), r = 1..(k-1). S_2 is A025480; S_3 gives: a(3*n) = n, a(3*n + 1) = a(2*n), a(3*n + 2) = a(2*n + 1), which is A263390.
The subsequence of all nonzero terms is A131987. (End)
Similar to but different from A108202. - N. J. A. Sloane, Nov 26 2020
This sequence can be otherwise defined in two alternative (but related) ways, with a(0)=0, as follows: (i) If a(n) is a novel term, then a(n+1) = a(a(n)); if a(n) has been seen before, most recently at a(m), then a(n+1) = n-m (as in A181391). (ii) As above for novel a(n), then if a(n) has been seen before, a(n+1) = smallest k < a(n) which is not already a term. - David James Sycamore, Jul 13 2021
From a binary perspective, the sequence can be seen as even,odd pairs where the odd value is the previous even value, dropping the rightmost bits up to and including the lowest zero bit, aka right-shifted past the lowest clear bit. E.g., (5)101 -> 1, (17)10001 -> (4)100, (29)11101 -> (7)111, (39)100111 -> (2)10. - Joe Nellis, Oct 09 2022

			From Deuard Worthen (deuard(AT)raytheon.com), Jan 27 2006: (Start)
The sequence can be constructed as a triangle as:
  0
  0  1
  0  2  1  3
  0  4  2  5  1  6  3  7
  0  8  4  9  2 10  5 11  1 12  6 13  3 14  7 15
  ...
At each stage we interleave the next 2^m numbers in the previous row. (End)
Left=0/1, Right=1/0: StB=A007305/A047679; Left=0/1, Right=1/1: StB=A007305/A007306; Left=1/3, Right=2/3: StB=A153161/A153162. - _Reinhard Zumkeller_, Dec 22 2008

L. Levine, Fractal sequences and restricted Nim, Ars Combin. 80 (2006), 113-127.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Josef Eschgfäller and Andrea Scarpante, Dichotomic random number generators, arXiv:1603.08500 [math.CO], 2016.
Ralf Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 3.2.4.
L. Levine, Fractal sequences and restricted Nim, arXiv:math/0409408 [math.CO], 2004.
Ralf Stephan, Some divide-and-conquer sequences ....
Ralf Stephan, Table of generating functions.
Paul Tarau, A Groupoid of Isomorphic Data Transformations, Calculemus 2009, 8th International Conference, MKM 2009, pp. 170-185, Springer, LNAI 5625.
Index entries for sequences related to the Josephus Problem.

The Y-projection of A075300.

Cf. A108202, A138002, A000265, A003602, A103391, A153733, A220466, A225381, A131987, A263390, A181391, A007814.

import Data.List
interleave xs ys = concat . transpose $ [xs,ys]
a025480 = interleave [0..] a025480
-- Cale Gibbard, Nov 18 2009

Cf. comments by Worthen and Hasler.
import Data.List (transpose)
a025480 n k = a025480_tabf !! n !! k
a025480_row n = a025480_tabf !! n
a025480_tabf = iterate (\xs -> concat $
   transpose [xs, [length xs .. 2 * length xs - 1]]) [0]
a025480_list = concat $ a025480_tabf
-- Reinhard Zumkeller, Apr 29 2012

A025480 := proc(n)
    option remember ;
    if type(n,'even') then
        n/2 ;
    else
        procname((n-1)/2) ;
    end if;
end proc:
seq(A025480(n),n=0..100) ; # R. J. Mathar, Jul 16 2020

a[n_] := a[n] = If[OddQ@n, a[(n - 1)/2], n/2]; Table[ a[n], {n, 0, 83}] (* Robert G. Wilson v, Mar 30 2006 *)
Table[BitShiftRight[n, IntegerExponent[n, 2] + 1], {n, 100}] (* IWABUCHI Yu(u)ki, Oct 13 2012 *)

a(n)={while(n%2,n\=2);n\2} \\ M. F. Hasler, May 03 2008

A025480(n)=n>>valuation(n*2+2,2) \\ M. F. Hasler, Apr 12 2012

def A025480(n): return n>>((~(n+1)&n).bit_length()+1) # Chai Wah Wu, Jul 13 2022

A025480 = lambda n: odd_part(n+1)//2
[A025480(n) for n in (0..83)] # Peter Luschny, May 20 2014

a(n) = A003602(n+1) - 1. [Corrected by Max Alekseyev, May 05 2022]
a(n) = (A000265(n+1)-1)/2 = ((n+1)/A006519(n+1)-1)/2.
a(n) = A153733(n)/2. - Reinhard Zumkeller, Dec 31 2008
2^A007814(n+1)*(2*a(n)+1) = n+1. (See functions hd, tl and cons in [Paul Tarau 2009].) - Paul Tarau (paul.tarau(AT)gmail.com), Mar 21 2010
a(3*n + 1) = A173732(n). - Reinhard Zumkeller, Apr 29 2012
a((2*n+1)*2^p-1) = n, p >= 0 and n >= 0. - Johannes W. Meijer, Jan 24 2013
a(n) = n - A225381(n). - Marcus Hedbring, May 18 2013
G.f.: -1/(1-x) + Sum_{k>=0} x^(2^k-1)/(1-2*x^2^(k+1)+x^2^(k+2)). - Ralf Stephan, May 19 2013
a(n) = A049084(A181363(n+1)). - Reinhard Zumkeller, Mar 22 2014
a(n) = floor(n / 2^A001511(n+1)). - Adam Shelly, Mar 05 2019
Recursion: a(0) = 0; a(n + 1) = a(a(n)) if a(n) is a first occurrence of a term, else a(n + 1) = n - a(n-1). - David James Sycamore, Apr 29 2020
a(n) * 2^(A007814(n+1)+1) + 2^A007814(n+1) - 1 = n (equivalent to the formula given in the comment by Paul Tarau). - Ruud H.G. van Tol, Apr 14 2023
Sum_{k=1..n} a(k) = n^2/6 + O(n). - Amiram Eldar, Aug 07 2023

Edited by M. F. Hasler, Mar 16 2018

A035327 Write n in binary, interchange 0's and 1's, convert back to decimal.

Original entry on oeis.org

1, 0, 1, 0, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46

Offset: 0

N. J. A. Sloane

For n>0: largest m<=n such that no carry occurs when adding m to n in binary arithmetic: A003817(n+1) = a(n) + n = a(n) XOR n. - Reinhard Zumkeller, Nov 14 2009
a(0) could be considered to be 0 (it was set so from 2004 to 2008) if the binary representation of zero was chosen to be the empty string. - Jason Kimberley, Sep 19 2011
For n > 0: A240857(n,a(n)) = 0. - Reinhard Zumkeller, Apr 14 2014
This is a base-2 analog of A048379. Another variant, without converting back to decimal, is given in A256078. - M. F. Hasler, Mar 22 2015
For n >= 2, a(n) is the least nonnegative k that must be added to n+1 to make a power of 2. Hence in a single-elimination tennis tournament with n entrants, a(n-1) is the number of players given a bye in round one, so that the number of players remaining at the start of round two is a power of 2. For example, if 39 players register, a(38)=25 players receive a round-one bye leaving 14 to play, so that round two will have 25+(14/2)=32 players. - Mathew Englander, Jan 20 2024

			8 = 1000 -> 0111 = 111 = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences related to binary expansion of n
Index entries for sequences related to the Josephus Problem

a(n) = A003817(n) - n, for n>0.
Cf. A080079, A087734, A167831, A167877, A007088, A061601, A171960, A010078, A000225, A006257 (Josephus problem).
Cf. A240857.
Cf. A048379, A256078.

a035327 n = if n <= 1 then 1 - n else 2 * a035327 n' + 1 - b
            where (n',b) = divMod n 2
-- Reinhard Zumkeller, Feb 21 2014

using IntegerSequences
A035327List(len) = [Bits("NAND", n, n) for n in 0:len]
println(A035327List(100))  # Peter Luschny, Sep 25 2021

A035327:=func; // Jason Kimberley, Sep 19 2011

seq(2^(1 + ilog2(max(n, 1))) - 1 - n, n = 0..81); # Emeric Deutsch, Oct 19 2008
A035327 := n -> `if`(n=0, 1, Bits:-Nand(n, n)):
seq(A035327(n), n=0..81); # Peter Luschny, Sep 23 2019

Table[BaseForm[FromDigits[(IntegerDigits[i, 2]/.{0->1, 1->0}), 2], 10], {i, 0, 90}]
Table[BitXor[n, 2^IntegerPart[Log[2, n] + 1] - 1], {n, 100}] (* Alonso del Arte, Jan 14 2006 *)
Join[{1},Table[2^BitLength[n]-n-1,{n,100}]] (* Paolo Xausa, Oct 13 2023 *)
Table[FromDigits[IntegerDigits[n,2]/.{0->1,1->0},2],{n,0,90}] (* Harvey P. Dale, May 03 2025 *)

a(n)=sum(k=1,n,if(bitxor(n,k)>n,1,0)) \\ Paul D. Hanna, Jan 21 2006

a(n) = bitxor(n, 2^(1+logint(max(n,1), 2))-1) \\ Rémy Sigrist, Jan 04 2019

a(n)=if(n, bitneg(n, exponent(n)+1), 1) \\ Charles R Greathouse IV, Apr 13 2020

def a(n): return int(''.join('1' if i == '0' else '0' for i in bin(n)[2:]), 2) # Indranil Ghosh, Apr 29 2017

def a(n): return 1 if n == 0 else n^((1 << n.bit_length()) - 1)
print([a(n) for n in range(100)]) # Michael S. Branicky, Sep 28 2021

def A035327(n): return (~n)^(-1<Chai Wah Wu, Dec 20 2022

def a(n):
    if n == 0:
        return 1
    return sum([(1 - b) << s for (s, b) in enumerate(n.bits())])
[a(n) for n in srange(82)]  # Peter Luschny, Aug 31 2019

a(n) = 2^k - n - 1, where 2^(k-1) <= n < 2^k.
a(n+1) = (a(n)+n) mod (n+1); a(0) = 1. - Reinhard Zumkeller, Jul 22 2002
G.f.: 1 + 1/(1-x)*Sum_{k>=0} 2^k*x^2^(k+1)/(1+x^2^k). - Ralf Stephan, May 06 2003
a(0) = 0, a(2n+1) = 2*a(n), a(2n) = 2*a(n) + 1. - Philippe Deléham, Feb 29 2004
a(n) = number of positive integers k < n such that n XOR k > n. a(n) = n - A006257(n). - Paul D. Hanna, Jan 21 2006
a(n) = 2^{1+floor(log[2](n))}-n-1 for n>=1; a(0)=1. - Emeric Deutsch, Oct 19 2008
a(n) = if n<2 then 1 - n else 2*a(floor(n/2)) + 1 - n mod 2. - Reinhard Zumkeller, Jan 20 2010
a(n) = abs(2*A053644(n) - n - 1). - Mathew Englander, Jan 22 2024

More terms from Vit Planocka (planocka(AT)mistral.cz), Feb 01 2003
a(0) corrected by Paolo P. Lava, Oct 22 2007
Definition completed by M. F. Hasler, Mar 22 2015

A038712 Let k be the exponent of highest power of 2 dividing n (A007814); a(n) = 2^(k+1)-1.

Original entry on oeis.org

1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 31, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 63, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 31, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 127, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 31, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 63, 1, 3

Offset: 1

Henry Bottomley, May 02 2000

n XOR n-1, i.e., nim-sum of a pair of consecutive numbers.
Denominator of quotient sigma(2*n)/sigma(n). - Labos Elemer, Nov 04 2003
a(n) = the Towers of Hanoi disc moved at the n-th move, using standard moves with discs labeled (1, 3, 7, 15, ...) starting from top (smallest = 1). - Gary W. Adamson, Oct 26 2009
Equals row sums of triangle A168312. - Gary W. Adamson, Nov 22 2009
In the binary expansion of n, delete everything left of the rightmost 1 bit, and set all bits to the right of it. - Ralf Stephan, Aug 22 2013
Every finite sequence of positive integers summing to n may be termwise dominated by a subsequence of the first n values in this sequence [see Bannister et al., 2013]. - David Eppstein, Aug 31 2013
Sum of powers of 2 dividing n. - Omar E. Pol, Aug 18 2019
Given the binary expansion of (n-1) as {b[k-1], b[k-2], ..., b[2], b[1], b[0]}, then the binary expansion of a(n) is {bitand(b[k-1], b[k-2], ..., b[2], b[1], b[0]), bitand(b[k-2], ..., b[2], b[1], b[0]), ..., bitand(b[2], b[1], b[0]), bitand(b[1], b[0]), b[0], 1}. Recursively stated - 0th bit (L.S.B) of a(n), a(n)[0] = 1, a(n)[i] = bitand(a(n)[i-1], (n-1)[i-1]), where n[i] = i-th bit in the binary expansion of n. - Chinmaya Dash, Jun 27 2020

			a(6) = 3 because 110 XOR 101 = 11 base 2 = 3.
From _Omar E. Pol_, Aug 18 2019: (Start)
Illustration of initial terms:
a(n) is also the area of the n-th region of an infinite diagram of compositions (ordered partitions) of the positive integers, where the length of the n-th horizontal line segment is equal to A001511(n) and the length of the n-th vertical line segment is equal to A006519(n), as shown below (first eight regions):
-----------------------------
n    a(n)    Diagram
-----------------------------
.            _ _ _ _
1     1     |_| | | |
2     3     |_ _| | |
3     1     |_|   | |
4     7     |_ _ _| |
5     1     |_| |   |
6     3     |_ _|   |
7     1     |_|     |
8    15     |_ _ _ _|
.
The above diagram represents the eight compositions of 4: [1,1,1,1],[2,1,1],[1,2,1],[3,1],[1,1,2],[2,2],[1,3],[4].
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. J. Bannister, Z. Cheng, W. E. Devanny, and D. Eppstein, Superpatterns and universal point sets, 21st Int. Symp. Graph Drawing, 2013, arXiv:1308.0403 [cs.CG], 2013.
Klaus Brockhaus, Illustration of A038712 and A080277.
David Eppstein, 1317131 and majorization by subsequences.
Fabrizio Frati, M. Patrignani, and V. Roselli, LR-Drawings of Ordered Rooted Binary Trees and Near-Linear Area Drawings of Outerplanar Graphs, arXiv preprint arXiv:1610.02841 [cs.CG], 2016.
Malgorzata J. Krawczyk, Paweł Oświęcimka, and Krzysztof Kułakowski, Ordered Avalanches on the Bethe Lattice, Entropy, Vol. 21, (2019) 968.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions.
Ralf Stephan, Table of generating functions.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Index entries for sequences related to binary expansion of n.
Index entries for sequences related to Nim-sums.

A038713 translated from binary, diagonals of A003987 on either side of main diagonal.
Cf. A062383. Partial sums give A080277.
Bisection of A089312. Cf. A088837.
a(n)-1 is exponent of 2 in A089893(n).
Cf. A130093.
This is Guy Steele's sequence GS(6, 2) (see A135416).
Cf. A001620, A168312, A220466, A361019 (Dirichlet inverse).

int a(int n) { return n ^ (n-1); } // Russ Cox, May 15 2007

import Data.Bits (xor)
a038712 n = n `xor` (n - 1) :: Integer  -- Reinhard Zumkeller, Apr 23 2012

nmax:=98: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 2^(p+1)-1 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 01 2013
# second Maple program:
a:= n-> Bits[Xor](n, n-1):
seq(a(n), n=1..98);  # Alois P. Heinz, Feb 02 2023

Table[Denominator[DivisorSigma[1, 2*n]/DivisorSigma[1, n]], {n, 1, 128}]
Table[BitXor[(n + 1), n], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

vector(66,n,bitxor(n-1,n)) \\ Joerg Arndt, Sep 01 2013; corrected by Michel Marcus, Aug 02 2018

A038712(n) = ((1<<(1+valuation(n,2)))-1); \\ Antti Karttunen, Nov 24 2024

def A038712(n): return n^(n-1) # Chai Wah Wu, Jul 05 2022

a(n) = A110654(n-1) XOR A008619(n). - Reinhard Zumkeller, Feb 05 2007
a(n) = 2^A001511(n) - 1 = 2*A006519(n) - 1 = 2^(A007814(n)+1) - 1.
Multiplicative with a(2^e) = 2^(e+1)-1, a(p^e) = 1, p > 2. - Vladeta Jovovic, Nov 06 2001; corrected by Jianing Song, Aug 03 2018
Sum_{n>0} a(n)*x^n/(1+x^n) = Sum_{n>0} x^n/(1-x^n). Inverse Moebius transform of A048298. - Vladeta Jovovic, Jan 02 2003
From Ralf Stephan, Jun 15 2003: (Start)
G.f.: Sum_{k>=0} 2^k*x^2^k/(1 - x^2^k).
a(2*n+1) = 1, a(2*n) = 2*a(n)+1. (End)
Equals A130093 * [1, 2, 3, ...]. - Gary W. Adamson, May 13 2007
Sum_{i=1..n} (-1)^A000120(n-i)*a(i) = (-1)^(A000120(n)-1)*n. - Vladimir Shevelev, Mar 17 2009
Dirichlet g.f.: zeta(s)/(1 - 2^(1-s)). - R. J. Mathar, Mar 10 2011
a(n) = A086799(2*n) - 2*n. - Reinhard Zumkeller, Aug 07 2011
a((2*n-1)*2^p) = 2^(p+1)-1, p >= 0. - Johannes W. Meijer, Feb 01 2013
a(n) = A000225(A001511(n)). - Omar E. Pol, Aug 31 2013
a(n) = A000203(n)/A000593(n). - Ivan N. Ianakiev and Omar E. Pol, Dec 14 2017
L.g.f.: -log(Product_{k>=0} (1 - x^(2^k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 15 2018
a(n) = 2^(1 + (A183063(n)/A001227(n))) - 1. - Omar E. Pol, Nov 06 2018
a(n) = sigma(n)/(sigma(2*n) - 2*sigma(n)) = 3*sigma(n)/(sigma(4*n) - 4*sigma(n)) = 7*sigma(n)/(sigma(8*n) - 8*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
Sum_{k=1..n} a(k) ~ n*log_2(n) + (1/2 + (gamma - 1)/log(2))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 24 2022
a(n) = Sum_{d divides n} m(d)*phi(d), where m(n) = Sum_{d divides n} (-1)^(d+1)* mobius(d). - Peter Bala, Jan 23 2024

Definition corrected by N. J. A. Sloane, Sep 07 2015 at the suggestion of Marc LeBrun

Name corrected by Wolfdieter Lang, Aug 30 2016

A062383 a(0) = 1: for n>0, a(n) = 2^floor(log_2(n)+1) or a(n) = 2*a(floor(n/2)).

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128

Offset: 0

Antti Karttunen, Jun 19 2001

Informally, write down 1 followed by 2^k 2^(k-1) times, for k = 1,2,3,4,... These are the denominators of the binary van der Corput sequence (see A030101 for the numerators). - N. J. A. Sloane, Dec 01 2019
a(n) is the denominator of the form 2^k needed to make the ratio (2n-1)/2^k lie in the interval [1-2], i.e. such ratios are 1/1, 3/2, 5/4, 7/4, 9/8, 11/8, 13/8, 15/8, 17/16, 19/16, 21/16, ... where the numerators are A005408 (The odd numbers).
Let A_n be the upper triangular matrix in the group GL(n,2) that has zero entries below the diagonal and 1 elsewhere. For example for n=4 the matrix is / 1,1,1,1 / 0,1,1,1 / 0,0,1,1 / 0,0,0,1 /. The order of this matrix as an element of GL(n,2) is a(n-1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 14 2001
A006257(n)/a(n) = (0, 0.1, 0.01, 0.11, 0.001, ...) enumerates all binary fractions in the unit interval [0, 1). - Fredrik Johansson, Aug 14 2006
a(n) = maximum of row n+1 in A240769. - Reinhard Zumkeller, Apr 13 2014
This is the discriminator sequence for the odious numbers. - N. J. A. Sloane, May 10 2016
From Jianing Song, Jul 05 2025: (Start)
a(n) is the period of {binomial(N,n) mod 2: N in Z}. For the general result, see A349593.
Since the modulus (2) is a prime, the remainder of binomial(N,n) is given by Lucas's theorem. (End)

Harry J. Smith, Table of n, a(n) for n = 0..1000
L. K. Arnold, S. J. Benkoski and B. J. McCabe, The discriminator (a simple application of Bertrand's postulate). Amer. Math. Monthly 92 (1985), 275-277.
Sajed Haque, Chapter 2.6.1 of Discriminators of Integer Sequences, 2017, See p. 33.
S. Haque and J. Shallit, Discriminators and k-regular sequences, arXiv:1605.00092 [cs.DM], 2016.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Wikipedia, Lucas's theorem
Index to divisibility sequences

Apart from the initial term, equals 2 * A053644. MASKTRANSi(A062383) seems to give a signed form of A038712. (See identities at A053644). floor_log_2 given in A054429.
Equals A003817(n)+1. Cf. A002884.
Bisection of A065285. Cf. A076877.
Equals for n>=1 the r(n) sequence of A160464. - Johannes W. Meijer, May 24 2009
Equals the r(n) sequence of A162440 for n>=1. - Johannes W. Meijer, Jul 06 2009
Cf. A030101, A264618, A264619.
Discriminator of the odious numbers (A000069). - Jeffrey Shallit, May 08 2016
Column 2 of A349593. A064235 (if offset 0), A385552, A385553, and A385554 are respectively columns 3, 5, 6, and 10.

import Data.List (transpose)
a062383 n = a062383_list !! n
a062383_list = 1 : zs where
   zs = 2 : (map (* 2) $ concat $ transpose [zs, zs])
-- Reinhard Zumkeller, Aug 27 2014, Mar 13 2014

[2^Floor(Log(2,2*n+1)): n in [0..70]]; // Bruno Berselli, Mar 04 2016

[seq(2^(floor_log_2(j)+1),j=0..127)]; or [seq(coerce1st_octave((2*j)+1),j=0..127)]; or [seq(a(j),j=0..127)];
coerce1st_octave := proc(r) option remember; if(r < 1) then coerce1st_octave(2*r); else if(r >= 2) then coerce1st_octave(r/2); else (r); fi; fi; end;
A062383 := proc(n)
    option remember;
    if n = 0 then
        1 ;
    else
        2*procname(floor(n/2));
    end if;
end proc:
A062383 := n -> 1 + Bits:-Iff(n, n):
seq(A062383(n), n=0..69); # Peter Luschny, Sep 23 2019

a[n_] := a[n] = 2 a[n/2 // Floor]; a[0] = 1; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 04 2016 *)
Table[2^Floor[Log2[n] + 1], {n, 0, 20}] (* Eric W. Weisstein, Nov 17 2017 *)
2^Floor[Log2[Range[0, 20]] + 1] (* Eric W. Weisstein, Nov 17 2017 *)
2^BitLength[Range[0, 100]] (* Paolo Xausa, Jan 29 2025 *)

{ a=1; for (n=0, 1000, write("b062383.txt", n, " ", a*=ceil((n + 1)/a)) ) } \\ Harry J. Smith, Aug 06 2009

a(n)=1<<(log(2*n+1)\log(2)) \\ Charles R Greathouse IV, Dec 08 2011

def A062383(n): return 1 << n.bit_length() # Chai Wah Wu, Jun 30 2022

a(1) = 1 and a(n+1) = a(n)*ceiling(n/a(n)). - Benoit Cloitre, Aug 17 2002
G.f.: 1/(1-x) * (1 + Sum_{k>=0} 2^k*x^2^k). - Ralf Stephan, Apr 18 2003
a(n) = A142151(2*n)/2 + 1. - Reinhard Zumkeller, Jul 15 2008
log(a(n))/log(2) = A029837(n+1). - Johannes W. Meijer, Jul 06 2009
a(n+1) = a(n) + A099894(n). - Reinhard Zumkeller, Aug 06 2009
a(n) = A264619(n) - A264618(n). - Reinhard Zumkeller, Dec 01 2015
a(n) is the smallest power of 2 > n. - Chai Wah Wu, Nov 04 2016
a(n) = 2^ceiling(log_2(n+1)). - M. F. Hasler, Sep 20 2017

A048724 Write n and 2n in binary and add them mod 2.

Original entry on oeis.org

0, 3, 6, 5, 12, 15, 10, 9, 24, 27, 30, 29, 20, 23, 18, 17, 48, 51, 54, 53, 60, 63, 58, 57, 40, 43, 46, 45, 36, 39, 34, 33, 96, 99, 102, 101, 108, 111, 106, 105, 120, 123, 126, 125, 116, 119, 114, 113, 80, 83, 86, 85, 92, 95, 90, 89, 72, 75, 78, 77, 68, 71, 66, 65, 192

Offset: 0

Antti Karttunen, Apr 26 1999

Reversing binary representation of -n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives -n. Note that the alternation is applied only to the nonzero bits and does not depend on the exponent of two. All integers have a unique reversing binary representation (see cited exercise for proof). Complement of A065621. - Marc LeBrun, Nov 07 2001
A permutation of the "evil" numbers A001969. - Marc LeBrun, Nov 07 2001
A048725(n) = a(a(n)). - Reinhard Zumkeller, Nov 12 2004

			12 = 1100 in binary, 24=11000 and their sum is 10100=20, so a(12)=20.
a(4) = 12 = + 8 + 4 -> - 8 + 4 = -4.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 178, (exercise 4.1. Nr. 27)

T. D. Noe, Table of n, a(n) for n = 0..1023
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
H. D. Nguyen, A mixing of Prouhet-Thue-Morse sequences and Rademacher functions, 2014. See Example 20. - _N. J. A. Sloane_, May 24 2014
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A048720, A048725, A048726, A048728.
Bisection of A003188 (even part).
See also A065620, A065621.
Cf. A242399.

import Data.Bits (xor, shiftL)
a048724 n = n `xor` shiftL n 1 :: Integer
-- Reinhard Zumkeller, Mar 06 2013

a:= n-> Bits[Xor](n, n+n):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 06 2016

Table[ BitXor[2n, n], {n, 0, 65}] (* Robert G. Wilson v, Jul 06 2006 *)

a(n)=bitxor(n,2*n) \\ Charles R Greathouse IV, Jan 04 2013

def A048724(n): return n^(n<<1) # Chai Wah Wu, Apr 05 2021

a(n) = Xmult(n, 3) (or n XOR (n<<1)).
a(n) = A065621(-n).
a(2n) = 2a(n), a(2n+1) = 2a(n) + 2(-1)^n + 1.
G.f. 1/(1-x) * sum(k>=0, 2^k*(3t-t^3)/(1+t)/(1+t^2), t=x^2^k). - Ralf Stephan, Sep 08 2003
a(n) = sum(k=0, n, (1-(-1)^round(+n/2^k))/2*2^k). - Benoit Cloitre, Apr 27 2005
a(n) = A001969(A003188(n)). - Philippe Deléham, Apr 29 2005
a(n) = A106409(2*n) for n>0. - Reinhard Zumkeller, May 02 2005
a(n) = A142149(2*n). - Reinhard Zumkeller, Jul 15 2008

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)

cp's OEIS Frontend

A069010 Number of runs of 1's in the binary representation of n.

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A079000 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is odd".

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A035263 Trajectory of 1 under the morphism 0 -> 11, 1 -> 10; parity of 2-adic valuation of 2n: a(n) = A000035(A001511(n)).

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

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A025480 a(2n) = n, a(2n+1) = a(n).

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