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A005599 Running sum of every third term in the {+1,-1}-version of Thue-Morse sequence A010060.
(Formerly M0468)
4
0, 1, 2, 3, 4, 5, 6, 7, 6, 7, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 16, 17, 18, 19, 18, 19, 20, 21, 20, 19, 18, 19, 18, 19, 20, 21, 22, 23, 24, 25, 24, 25, 26, 27, 28, 29, 30, 29, 30, 31, 32, 33, 34, 35, 36, 35, 36, 35, 34, 33, 34, 35, 36, 35, 36, 37, 38, 39, 40, 41, 42, 43, 42, 43, 44, 45, 46, 47, 48, 47, 48, 49, 50 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 98.

Hofer, Roswitha. "Coquet-type formulas for the rarefied weighted Thue-Morse sequence." Discrete Mathematics 311.16 (2011): 1724-1734.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

J. Coquet, A summation formula related to the binary digits, Invent. Math. 73 (1983) 107-115.

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

P. J. Grabner and H.-K. Hwang, Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence, Constructive Approximation, Jan. 2005, Volume 21, Issue 2, pp 149-179.

D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc., 21 (1969), 719-721.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

M. R. Schroeder & N. J. A. Sloane, Correspondence, 1991

FORMULA

a(n) = Sum( (-1)^wt(3*k),k=0..n-1). See Allouche-Shallit for asymptotics. - From N. J. A. Sloane, Jul 22 2012

The generating function -(2*z**4+z**3+z+1)*(z**3-z**2-1)/(z**6+z**5+z**4+z**3+z**2+z+1)/(z-1)**2 proposed in the Plouffe thesis is wrong.

MAPLE

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;

f:=n->add( (-1)^wt(3*k), k=0..n-1);

[seq(f(n), n=0..50)]; # N. J. A. Sloane, Jul 22 2012

A005599 := proc(n)

        add( A106400(3*i), i=0..n-1) ;

end proc: # R. J. Mathar, Jul 22 2012

MATHEMATICA

wt[n_] := DigitCount[n, 2, 1]; a[n_] := Sum[(-1)^wt[3*k], {k, 0, n-1}]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 03 2014, after N. J. A. Sloane *)

PROG

(Haskell)

a005599 n = a005599_list !! n

a005599_list = scanl (+) 0 $ f a106400_list

   where f (x:_:_:xs) = x : f xs

-- Reinhard Zumkeller, May 26 2013

(PARI) a(n) = sum(k=0, n-1, (-1)^hammingweight(3*k)); \\ Michel Marcus, Jul 03 2017

CROSSREFS

See A000120 for "wt" (the binary weight of n).

Cf. A010060, A106400.

Sequence in context: A161209 A279513 A000026 * A071934 A161658 A066853

Adjacent sequences:  A005596 A005597 A005598 * A005600 A005601 A005602

KEYWORD

nonn,easy,nice

AUTHOR

M. R. Schroeder

STATUS

approved

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)