0,4

a(n) is also the largest integer such that 2^a(n) divides binomial(2n,n)=A000984(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 27 2002

To construct the sequence, start with 0 and use the rule: If k>=0 and a(0),a(1),...,a(2^k-1) are the 2^k first terms, then the next 2^k terms are a(0)+1,a(1)+1,...,a(2^k-1)+1. - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 30 2003

An example of a fractal sequence. That is, if you omit every other number in the sequence, you get the original sequence. And of course this can be repeated. So if you form the sequence a(0 * 2^n), a(1 * 2^n), a(2 * 2^n), a(3 * 2^n), ... (for any integer n > 0), you get the original sequence. - Christopher.Hills(AT)sepura.co.uk, May 14, 2003

The n-th row of Pascal's triangle has 2^k distinct odd binomial coefficients where k=a(n)-1. - Lekraj Beedassy (blekraj(AT)yahoo.com), May 15 2003

Fixed point of the morphism 0 -> 01, 1 -> 12, 2 -> 23, 3 -> 34, 4 -> 45, etc., starting from a(0) = 0. - Robert G. Wilson v, Jan 24 2006. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Jan 25 2006

a(n) = number of times n appears among the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Jan 25 2006

a(n) = number of solutions of the Diophantine equation 2^m*k+2^(m-1)+i=n, where m>=1, k>=0, 0<=i<2^(m-1); a(5)=2 because only (m,k,i)=(1,2,0) [2^1*2+2^0+0=5] and (m,k,i)=(3,0,1) [2^3*0+2^2+1=5] are solutions. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 31 2006

The first appearance of k, k=>0, is at a(2^k-1). - Robert G. Wilson v Jul 27 2006

a(n) = A138530(n,2) for n > 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 26 2008

Sequence is given by T^(infty)(0) where T is the operator transforming any word w=w(1)w(2)...w(m) into T(w)=w(1)(w(1)+1)w(2)(w(2)+1)...w(m)(w(m)+1). i.e. T(0)=01, T(01)=0112, T(0112)=01121223. [From Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 04 2009]

a(A077436(n))=A159918(A077436(n)); a(A000290(n))=A159918(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 25 2009]

For n>=2, the minimal k for which a(k(2^n-1)) is not multiple of n is 2^n+3. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jun 05 2009]

a(n) = A063787(n) - A007814(n). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 2009]

Triangle inequality: a(k+m)<=a(k)+a(m). Equality holds iff C(k+m,m) is odd. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jul 19 2009]

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340. [From N. J. A. Sloane, Mar 12 2009]

Emmanuel Ferrand, Deformations of the Taylor Formula, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7.

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.

M. D. McIlroy, The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput., 3 (1974), 255-261.

Problem B-82, Fib. Quart., 4 (1966), 374-375.

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).

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)

P. Flajolet et al., Mellin Transforms And Asymptotics: Digital Sums, Theoret. Computer Sci. 23 (1994), 291-314.

Michael Gilleland, Some Self-Similar Integer Sequences

Nick Hobson, Python program for this sequence

K. Q. Ji and H. S. Wilf, Extreme Palindromes

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Digit Sum

Wolfram Research, Numbers in Pascal's triangle

Index entries for "core" sequences

Index entries for sequences related to binary expansion of n

a(0) = 0, a(2n) = a(n), a(2n+1) = a(n) + 1.

a(0) = 0, a(2^i) = 1; otherwise if n = 2^i + j with 0 < j < 2^i, a(n) = a(j) + 1.

G.f.: Product_{k >= 0} (1 + y*x^(2^k)) = Sum_{n >= 0} y^a(n)*x^n. - N. J. A. Sloane, Jun 04 2009

a(n) = a(n-1)+1-A007814(n) = log2[A001316(n)] = 2n-A005187(n) = A070939(n)-A023416(n). - Henry Bottomley (se16(AT)btinternet.com), Apr 04 2001; corrected by Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 15 2002

a(n)=log2(A000984(n)/A001790(n) ). - Benoit Cloitre, Oct 02 2002

For n>0, a(n)=n-sum(k=1, n, A007814(k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 19 2002

a(n)=n-sum(k>0, floor(n/2^k))=n-A011371(n). - Benoit Cloitre, Dec 19 2002

G.f.: 1/(1-x) * Sum(k>=0, x^(2^k)/(1+x^(2^k))). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 19 2003

a(0)=0, a(n)=a(n-2^log_2(floor(n)))+1. Examples: a(6)=a(6-2^2)+1=a(2)+1=a(2-2^1)+1+1=a(0)+2=2; a(101)=a(101-2^6)+1=a(37)+1=a(37-2^5)+2=a(5-2^2)+3=a(1-2^0)+4=a(0)+4=4; a(6275)=a(6275-2^12)+1=a(2179-2^11)+2=a(131-2^7)+3=a(3-2^1)+4=a(1-2^0)+5=5; a(4129)=a(4129-2^12)+1=a(33-2^5)+2=a(1-2^0)+3=3; - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 22 2006

A fixed point of the mapping 0 -> 01, 1 -> 12, 2 -> 23, 3 -> 34, 4 -> 45, ... With f(i) = floor(n/2^i), a(n) is the number of odd numbers in the sequence f(0), f(1), f(2), f(3), f(4), f(5), ... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 04 2004

When read mod 2 gives the Morse-Thue sequence A010060.

Let floor_pow4(n) denote n rounded down to the next power of four, floor_pow4(n) = 4 ^ floor(log4 n). Then a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2, a(n) = a(floor(n / floor_pow4(n))) + a(n % floor_pow4(n)) - Stephen K. Touset (stephen(AT)touset.org), Apr 04 2007

a(n)=n-sum{2<=k<=n, sum{j|n,j>=2, floor(log_2(j))-floor(log_2(j-1))}}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007

a(n)=A007814(C(2n,n))=1+A007814(C(2n-1,n)). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jul 20 2009]

a(4) = a(0) + a(0) = 0

a(8) = a(2) + a(0) = 1

a(13) = a(3) + a(1) = 2 + 1 = 3

a(23) = a(1) + a(7) = 1 + a(1) + a(3) = 1 + 1 + 2 = 4

Gary Adamson points out (Jun 03 2009) that this can be written as a triangle:

.0,

.1,

.1,2,

.1,2,2,3,

.1,2,2,3,2,3,3,4,

.1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,

.1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,

.1,2,2,3,2,3,...

where the rows converge to A063787.

Contribution from Omar E. Pol (info(AT)polprimos.com), Jun 07 2009: (Start)

Also, triangle begins:

0;

1,1;

2,1,2,2;

3,1,2,2,3,2,3,3;

4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4;

5,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5;

6,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,...

(End)

A000120 := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120;

Table[ Count[ IntegerDigits[n, 2], 1], {n, 0, 100} ]

Nest[ Flatten[ #1 /. a_Integer -> {a, a + 1}] &, {0}, 7] (* Robert G. Wilson v Jul 27 2006 *)

Table[Plus @@ IntegerDigits[n, 2], {n, 0, 104}]

(PARI) a(n)=if(n<0, 0, 2*n-valuation((2*n)!, 2))

(PARI) a(n)=if(n<0, 0, subst(Pol(binary(n)), x, 1))

(PARI) a(n)=if(n<1, 0, a(n\2)+n%2) - Michael Somos Mar 06 2004

Common LISP: (defun floor-to-power (n pow) (declare (fixnum pow)) (expt pow (floor (log n pow)))) (defun enabled-bits (n) (if (< n 4) (nth n (list 0 1 1 2)) (+ (enabled-bits (floor (/ n (floor-to-power n 4)))) (enabled-bits (mod n (floor-to-power n 4)))))) - Stephen K. Touset (stephen(AT)touset.org), Apr 04 2007

See link in A139351 for Fortran program.

The basic sequences concerning the binary expansion of n are this one, A000788, A000069, A001969, A023416, A059015, A007088.

a(n)=n-A011371[n]. - Labos E. (labos(AT)ana.sote.hu), Jul 27 2000

Sum of digits of n written in base 2-16: this sequence, A053735, A053737, A053824, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.

Cf. A007953.

This is Guy Steele's sequence GS(3, 4) (see A135416).

Cf. A007814.

Cf. A000079. [From Omar E. Pol (info(AT)polprimos.com), Jun 07 2009]

Sequence in context: A105056 A105061 A105164 this_sequence A105062 A106487 A105102

Adjacent sequences: A000117 A000118 A000119 this_sequence A000121 A000122 A000123

nonn,easy,core,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Stephen K. Touset (stephen(AT)touset.org), Apr 04 2007