

A005811


Number of runs in binary expansion of n (n>0); number of 1's in Gray code for n.
(Formerly M0110)


204



0, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 3, 2, 1, 2, 3, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 5, 4, 3, 2, 3, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 5, 4, 3, 4, 5, 6, 5, 6, 7, 6, 5, 4, 5, 6, 5, 4, 5
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OFFSET

0,3


COMMENTS

Starting with a(1) = 0 mirror all initial 2^k segments and increase by one.
a(n) gives the net rotation (measured in right angles) after taking n steps along a dragon curve.  Christopher Hendrie (hendrie(AT)acm.org), Sep 11 2002
This sequence generates A082410: (0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, ...) and A014577; identical to the latter except starting 1, 1, 0, ...; by writing a "1" if a(n+1) > a(n); if not, write "0". E.g., A014577(2) = 0, since a(3) < a(2), or 1 < 2.  Gary W. Adamson, Sep 20 2003
Starting with 1 = partial sums of A034947: (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...).  Gary W. Adamson, Jul 23 2008
The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.
Can be used as a binomial transform operator: Let a(n) = the nth term in any S(n); then extract 2^k strings, adding the terms. This results in the binomial transform of S(n). Say S(n) = 1, 3, 5, ...; then we obtain the strings: (1), (3, 1), (3, 5, 3, 1), (3, 5, 7, 5, 3, 5, 3, 1), ...; = the binomial transform of (1, 3, 5, ...) = (1, 4, 12, 32, 80, ...). Example: the 8bit string has a sum of 32 with a distribution of (1, 3, 3, 1) or one 1, three 3's, three 5's, and one 7; as expected.  Gary W. Adamson, Jun 21 2012
Considers all positive odd numbers as nodes of a graph. Two nodes are connected if and only if the sum of the two corresponding odd numbers is a power of 2. Then a(n) is the distance between 2n + 1 and 1.  Jianing Song, Apr 20 2019


REFERENCES

Danielle Cox and K. McLellan, A problem on generation sets containing Fibonacci numbers, Fib. Quart., 55 (No. 2, 2017), 105113.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000
Joerg Arndt, Matters Computational (The Fxtbook)
J.P. Allouche, G.N. Han and J. Shallit, On some conjectures of P. Barry, arXiv:2006.08909 [math.NT], 2020.
J.P. Allouche and J. Shallit, The Ring of kregular Sequences, II
J.P. Allouche and J. Shallit, The ring of kregular sequences, II, Theoret. Computer Sci., 307 (2003), 329.
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves  I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 6681, and number 3, July 1970, pages 133149. Reprinted with addendum in Donald E. Knuth, Selected Papers on Fun and Games, 2010, pages 571614. Equation 3.2 g(n) = a(n1).
P. Flajolet et al., Mellin Transforms And Asymptotics: Digital Sums, Theoret. Computer Sci. 23 (1994), 291314.
P. Flajolet and Lyle Ramshaw, A note on Gray code and oddeven merge, SIAM J. Comput. 9 (1980), 142158.
S. Kropf and S. Wagner, qQuasiadditive functions, arXiv:1605.03654 [math.CO], 2016.
Sara Kropf and S. Wagner, On qQuasiadditive and qQuasimultiplicative Functions, arXiv preprint arXiv:1608.03700 [math.CO], 2016.
Shuo Li, Palindromic length sequence of the ruler sequence and of the perioddoubling sequence, arXiv:2007.08317 [math.CO], 2020.
Helmut Prodinger and Friedrich J. Urbanek, Infinite 01Sequences Without Long Adjacent Identical Blocks, Discrete Mathematics, volume 28, issue 3, 1979, pages 277289. Also first author's copy. Their "variation" v(k) at definition 3.4 is a(k).
Jeffrey Shallit, The mathematics of Per Noergaard's rhythmic infinity system, Fib. Q., 43 (2005), 262268.
Ralf Stephan, Some divideandconquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for "core" sequences
Index entries for sequences related to binary expansion of n


FORMULA

a(2^k + i) = a(2^k  i + 1) + 1 for k >= 0 and 0 < i <= 2^k.  Reinhard Zumkeller, Aug 14 2001
a(2n+1) = 2a(n)  a(2n) + 1, a(4n) = a(2n), a(4n+2) = 1 + a(2n+1).
a(j+1) = a(j) + (1)^A014707(j).  Christopher Hendrie (hendrie(AT)acm.org), Sep 11 2002
G.f.: (1/(1x)) * Sum_{k>=0} x^2^k/(1+x^2^(k+1)).  Ralf Stephan, May 02 2003
Delete the 0, make subsets of 2^n terms; and reverse the terms in each subset to generate A088696.  Gary W. Adamson, Oct 19 2003
a(0) = 0, a(2n) = a(n) + [n odd], a(2n+1) = a(n) + [n even].  Ralf Stephan, Oct 20 2003
a(n) = Sum_{k=1..n} (1)^((k/2^A007814(k)1)/2) = Sum_{k=1..n} (1)^A025480(k1).  Ralf Stephan, Oct 29 2003
a(n) = A069010(n) + A033264(n).  Ralf Stephan, Oct 29 2003
a(0) = 0 then a(n) = a(floor(n/2)) + (a(floor(n/2)) + n) mod 2.  Benoit Cloitre, Jan 20 2014
a(n) = A037834(n) + 1.


EXAMPLE

Considered as a triangle with 2^k terms per row, the first few rows are:
1
2, 1
2, 3, 2, 1
2, 3, 4, 3, 2, 3, 2, 1
... nth row becomes right half of next row; left half is mirrored terms of nth row increased by one.  Gary W. Adamson, Jun 20 2012


MAPLE

A005811 := proc(n)
local i, b, ans;
if n = 0 then
return 0 ;
end if;
ans := 1;
b := convert(n, base, 2);
for i from nops(b)1 to 1 by 1 do
if b[ i+1 ]<>b[ i ] then
ans := ans+1
fi
od;
return ans ;
end proc:
seq(A005811(i), i=1..50) ;


MATHEMATICA

Table[ Length[ Length/@Split[ IntegerDigits[ n, 2 ] ] ], {n, 1, 255} ]


PROG

(PARI) a(n)=sum(k=1, n, (1)^((k/2^valuation(k, 2)1)/2))
(PARI) a(n)=if(n<1, 0, a(n\2)+(a(n\2)+n)%2) \\ Benoit Cloitre, Jan 20 2014
(PARI) a(n) = hammingweight(bitxor(n, n>>1)); \\ Gheorghe Coserea, Sep 03 2015
(Haskell)
import Data.List (group)
a005811 0 = 0
a005811 n = length $ group $ a030308_row n
a005811_list = 0 : f [1] where
f (x:xs) = x : f (xs ++ [x + x `mod` 2, x + 1  x `mod` 2])
 Reinhard Zumkeller, Feb 16 2013, Mar 07 2011
(Python)
def a(n): return bin(n^(n>>1))[2:].count("1") # Indranil Ghosh, Apr 29 2017


CROSSREFS

Cf. A037834 (1), A088748 (+1), A246960 (mod 4), A034947 (first differences), A000975 (indices of record highs).
Cf. A056539, A014707, A014577, A082410, A030308, A090079, A044813, A165413, A226227, A226228, A226229.
Partial sums of A112347.
Sequence in context: A267108 A004738 A043554 * A008342 A277214 A278603
Adjacent sequences: A005808 A005809 A005810 * A005812 A005813 A005814


KEYWORD

easy,nonn,core,nice,hear


AUTHOR

N. J. A. Sloane, Jeffrey Shallit, Simon Plouffe


EXTENSIONS

Additional description from Wouter Meeussen


STATUS

approved



