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A106400
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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.
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10
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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
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OFFSET
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0,1
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COMMENTS
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a(A000069(n)) = -1; a(A001969(n)) = +1. [Reinhard Zumkeller, Apr 29 2012]
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LINKS
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_Reinhard Zumkeller_, Table of n, a(n) for n = 0..10000
Joerg Arndt, Fxtbook
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FORMULA
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a(n)=(-1)^A010060(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).
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MAPLE
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A106400 := proc(n)
1-2*A010060(n) ;
end proc: # R. J. Mathar, Jul 22 2012
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PROG
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(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
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CROSSREFS
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Convolution inverse of A018819.
Sequence in context: A112865 A114523 A130151 A143431 A064179 A065357 A121241
Adjacent sequences: A106397 A106398 A106399 * A106401 A106402 A106403
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KEYWORD
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sign
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AUTHOR
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Michael Somos, May 02 2005
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STATUS
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approved
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