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A106400 Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 1 and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 1's and -1's. 11
1, -1, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

See A010060, the main entry for the Thue-Morse sequence, for additional information. - N. J. A. Sloane, Aug 13 2014

a(A000069(n)) = -1; a(A001969(n)) = +1. - Reinhard Zumkeller, Apr 29 2012

Partial sums of every third terms give A005599. - Reinhard Zumkeller, May 26 2013

Fixed point of the morphism 1 --> 1,-1 and -1 --> -1,1. - Robert G. Wilson v, Apr 07 2014

REFERENCES

Yann Bugeaud and Guo-Niu Han,, A combinatorial proof of the non-vanishing of Hankel determinants of the Thue-Morse sequence, Electronic Journal of Combinatorics 21(3) (2014), #P3.26.

LINKS

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

Joerg Arndt, Matters Computational (The Fxtbook)

Philip Lafrance, Narad Rampersad, Randy Yee, Some properties of a Rudin-Shapiro-like sequence, arXiv:1408.2277 [math.CO], 2014.

FORMULA

a(n) = (-1)^A010060(n).

a(n) = (-1)^wt(n), where wt(n) is the binary weight of n, A000120(n).

G.f.: A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 - 2uvw + u^2w.

G.f.: A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u6*u1^3 - 3*u6*u2*u1^2 + 3*u6*u2^2*u1 - u3*u2^3.

Euler transform of sequence b(n) where b(2^k) = -1 and zero otherwise.

G.f.: Product_{k>=0} (1-x^(2^k)) = A(x) = (1-x)A(x^2).

MAPLE

A106400 := proc(n)

        1-2*A010060(n) ;

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

MATHEMATICA

tm[0] = 0; tm[n_?EvenQ] := tm[n/2]; tm[n_] := 1 - tm[(n-1)/2]; Table[(-1)^tm[n], {n, 0, 101}] (* Jean-François Alcover, Oct 24 2013 *)

Nest[ Flatten[# /. {1 -> {1, -1}, -1 -> {-1, 1}}] &, {1}, 7] (* Robert G. Wilson v, Apr 07 2014 *)

PROG

(PARI) {a(n)=if(n<1, n>=0, a(n\2)*(-1)^(n%2))}

(PARI) {a(n)=local(A, m); if(n<1, n==0, m=1; A=1+O(x); while(m<=n, m*=2; A=subst(A, x, x^2)*(1-x)); polcoeff(A, n))}

(Haskell)

import Data.List (transpose)

a106400 n = a106400_list !! n

a106400_list =  1 : concat

   (transpose [map negate a106400_list, tail a106400_list])

-- Reinhard Zumkeller, Apr 29 2012

CROSSREFS

Convolution inverse of A018819.

Cf. A010060 (0 -> 1 & 1 -> -1), A000120, A005599, A000069, A001969.

Sequence in context: A112865 A114523 A130151 A143431 A064179 A065357 A121241

Adjacent sequences:  A106397 A106398 A106399 * A106401 A106402 A106403

KEYWORD

sign

AUTHOR

Michael Somos, May 02 2005

STATUS

approved

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Last modified December 19 10:51 EST 2014. Contains 252207 sequences.