A005599 Running sum of every third term in the {+1,-1}-version of Thue-Morse sequence A010060. (Formerly M0468)

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

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; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 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 A337642 A331303

Adjacent sequences: A005596 A005597 A005598 * A005600 A005601 A005602

Keyword

nonn,easy,nice

Author

M. R. Schroeder

Status

approved