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A029883
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First differences of Thue-Morse sequence A001285.
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12
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1, 0, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 0, 1, 0, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 1, 0, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 1, 0, -1, 0, 1, 0, -1, 1, -1, 0, 1, 0, -1
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OFFSET
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1,1
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COMMENTS
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Fixed point of the morphism a->abc, b->ac, c->b, with a = 1, b = 0, c = -1, starting with a(1) = 1. - Philippe Deléham
This sequence, interpreted as an infinite word, is squarefree.
Let & represent concatenation. For a word w of integers, let -w be the same word with each symbol negated. Then, starting with the empty word, this sequence can be obtained by iteratively applying the transformation T(w) = w & 1 & -w & 0 & -w & -1 & w. (End)
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LINKS
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J.-P. Allouche and Jeffrey Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
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FORMULA
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Recurrence: a(4*n) = a(n), a(4*n+1) = a(2*n+1), a(4*n+2) = 0, a(4*n+3) = -a(2*n+1), starting a(1) = 1.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v+w+u^2-v^2+2*w^2-2*u*w. - Michael Somos, Jul 08 2004
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MATHEMATICA
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Nest[ Function[ l, {Flatten[(l /. {0 -> {1, -1}, 1 -> {1, 0, -1}, -1 -> {0}})]}], {1}, 7] (* Robert G. Wilson v, Feb 26 2005 *)
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PROG
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(PARI) a(n)=if(n<1||valuation(n, 2)%2, 0, -(-1)^subst(Pol(binary(n)), x, 1)) /* Michael Somos, Jul 08 2004 */
(Python)
def A029883(n): return (bin(n).count('1')&1)-(bin(n-1).count('1')&1) # Chai Wah Wu, Mar 03 2023
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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