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A000120
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1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n).
(Formerly M0105 N0041)
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588
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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, 3, 4, 4, 5, 3
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OFFSET
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0,4
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COMMENTS
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The binary weight of n is also called Hamming weight of n.
a(n) is also the largest integer such that 2^a(n) divides binomial(2n,n)=A000984(n) - Benoit Cloitre, 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, 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, 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, Jan 25 2006
a(n) = number of times n appears among the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421. - Jeremy Gardiner, 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, 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, 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, Mar 04 2009]
a(A077436(n))=A159918(A077436(n)); a(A000290(n))=A159918(n). [From Reinhard Zumkeller, 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, Jun 05 2009]
a(n) = A063787(n) - A007814(n). [From Gary W. Adamson, Jun 04 2009]
Triangle inequality: a(k+m)<=a(k)+a(m). Equality holds iff C(k+m,m) is odd. [From Vladimir Shevelev, Jul 19 2009]
Conjecture: The sequence where a(n) is the sum of digits of (n written in base b), can be written as a triangle T(r,k) in which all positive terms in column k are equal. Row 0 is a(0)=0 and row r lists a(b^(r-1)) .. a(b^r - 1), for r>=1 and b>=2. [From Omar E. Pol, Feb 20 2010]
The number of occurrences of value k in the first 2^n terms of A000120 is equal to the sum of the first n-k terms of the sequence T(k,i), where T(0,i) = 1,0,0,0,0,0,...(A000007), T(0,i) = 1,1,1,1,1,1,...(A000012), T(2,i) = 1,2,3,4,5,6,...(A000027), T(3,i) = 1,3,6,10,15,...(A000217), T(4,i) = 1,4,10,20,35...(A000292), and in general T(u,1) = 1; T(1,v) = 0 for v > 1; T(u,v) = T(u,v-1) + T(u-1, v) for u, v > 1 [From Brent Spillner (spillner(AT)acm.org), Sep 01 2010]
a(n) = A213629(n,1) for n > 0. - Reinhard Zumkeller, Jul 04 2012
Let m be the number of parts in the listing of the compositions of n as non-decreasing lists in lexicographic order, a(k) = n - length(composition(k)) for all k < 2^n and all n (see example); A007895 gives the equivalent for compositions into odd parts, A218829 gives the equivalent for integer partitions. [Joerg Arndt, Nov 09 2012]
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REFERENCES
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J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 119.
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.
D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.1.3, Problem 41, p. 589. - From N. J. A. Sloane, Aug 03 2012
M. D. McIlroy, The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput., 3 (1974), 255-261.
Sam Northshield, "Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,...", Amer. Math. Month., Vol. 117 (7), pp. 581-598, 2010.
Problem B-82, Fib. Quart., 4 (1966), 374-375.
C. Sanna, On Arithmetic Progressions of Integers with a Distinct Sum of Digits, Journal of Integer Sequences, Vol. 15 (2012), #12.8.1. - From N. J. A. Sloane, Dec 29 2012
Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W.H. Freeman, 1991, p. 383.
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).
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LINKS
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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
Robert Walker, Self Similar Sloth Canon Number Sequences
Eric Weisstein's World of Mathematics, Binary, Digit Count, Stolarsky-Harborth Constant, Digit Sum.
Wolfram Research, Numbers in Pascal's triangle
Index entries for "core" sequences
Index entries for sequences related to binary expansion of n
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FORMULA
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a(0) = 0, a(2*n) = a(n), a(2*n+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, Apr 04 2001; corrected by Ralf Stephan, 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, 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, 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, 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), ... - Philippe Deléham, 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, Jun 18 2007
a(n)=A007814(C(2n,n))=1+A007814(C(2n-1,n)). [From Vladimir Shevelev, Jul 20 2009]
For odd m>=1, a((4^m-1)/3)=a((2^m+1)/3)+(m-1)/2 (mod 2). [From Vladimir Shevelev, Sep 03 2010]
a(n) - a(n-1) = { 1 - a(n-1) iff A007814(n) = a(n-1), 1 iff A007814(n) = 0, -1 for all other A007814(n) } [From Brent Spillner (spillner(AT)acm.org), Sep 01 2010]
a(A001317(n))=2^a(n). [From Vladimir Shevelev, Oct 25 2010]
a(n)=A139351(n)+A139352(n)=Sum_k {A030308(n,k)}. - From Philippe Deléham, Oct 14 2011.
Contribution from Hieronymus Fischer, Jun 10 2012 (Start):
a(n) = sum_{j=1..m+1} (floor(n/2^j + 1/2)) - floor(n/2^j)), where m=floor(log_2(n)).
General formulas for the number of digits >= d in the base p representation of n, where 1 <= d < p.
a(n) = sum_{j=1..m+1} (floor(n/p^j + (p-d)/p) - floor(n/p^j)), where m=floor(log_p(n)).
G.f.: g(x) = (1/(1-x))*sum_{j>=0} (x^(d*p^j) - x^(p*p^j))/(1-x^(p*p^j)). (End)
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EXAMPLE
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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, 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)
From Joerg Arndt, Nov 09 2012: (Start)
Connection to the compositions of n as non-decreasing lists (see comment):
[ #]: a(n) composition
[ 0]: [0] 1 1 1 1 1
[ 1]: [1] 1 1 1 2
[ 2]: [1] 1 1 2 1
[ 3]: [2] 1 1 3
[ 4]: [1] 1 2 1 1
[ 5]: [2] 1 2 2
[ 6]: [2] 1 3 1
[ 7]: [3] 1 4
[ 8]: [1] 2 1 1 1
[ 9]: [2] 2 1 2
[10]: [2] 2 2 1
[11]: [3] 2 3
[12]: [2] 3 1 1
[13]: [3] 3 2
[14]: [3] 4 1
[15]: [4] 5
(End)
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MAPLE
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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;
A000120 := proc(n) add(i, i=convert(n, base, 2)) end: - P. Luschny, Feb 03 2011
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MATHEMATICA
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Table[ DigitCount[n, 2, 1], {n, 0, 105}]
Nest[ Flatten[# /. # -> {#, # + 1}] &, {0}, 7] (* Robert G. Wilson v, Sep 27 2011 *)
Table[Plus @@ IntegerDigits[n, 2], {n, 0, 104}]
Nest[Join[#, # + 1] &, {0}, 7] (* IWABUCHI Yu(u)ki, Jul 19 2012 *)
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PROG
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(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
(PARI) a(n)=my(v=binary(n)); sum(i=1, #v, v[i]) \\ Charles R Greathouse IV, Jun 24 2011
(PARI) A000120(n)=norml2(binary(n)) \\ - M. F. Hasler, Oct 09 2012
Common LISP: (defun floor-to-power (n pow) (declare (fixnum pow)) (expt pow (floor (log n pow)))) (defun enabled-bits (n) (if (< n 4) (n-th 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.
(Haskell)
a000120 0 = 0
a000120 n = a000120 n' + m where (n', m) = divMod n 2
a000120_list = 0 : c [1] where c (x:xs) = x : c (xs ++ [x, x+1])
-- Reinhard Zumkeller, Feb 19 2012, Jun 16 2011, Mar 07 2011
(Sage)
def A000120(n):
if n <= 1: return Integer(n)
return A000120(n//2) + n%2
[A000120(n) for n in range(105)] # Peter Luschny, Nov 19 2012
(Sage) def A000120(n) : return sum(n.digits(2)) # Eric M. Schmidt, Apr 26 2013
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CROSSREFS
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The basic sequences concerning the binary expansion of n are this one, A000788, A000069, A001969, A023416, A059015, A007088.
Partial sums see A000788.
0's counting sequence see A023416.
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. A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564 (for base 10).
Sequence in context: A105056 A105061 A105164 * A105062 A106487 A105102
Adjacent sequences: A000117 A000118 A000119 * A000121 A000122 A000123
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KEYWORD
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nonn,easy,core,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Stephen K. Touset (stephen(AT)touset.org), Apr 04 2007
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STATUS
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approved
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