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A020985
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The Rudin-Shapiro or Golay-Rudin-Shapiro sequence.
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12
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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
(list;
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OFFSET
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0,1
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COMMENTS
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Other names are the Rudin-Shapiro or Golay-Rudin-Shapiro infinite word.
Related to paper-folding sequences - see the Mendès France and Tenenbaum article.
a(A022155(n)) = -1; a(A203463(n)) = 1. [Reinhard Zumkeller, Jan 02 2012]
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REFERENCES
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J.-P. Allouche, Lecture notes on automatic sequences, Krakow October 2013; http://ssdnm.mimuw.edu.pl/pliki/wyklady/allouche-uj.pdf
J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 78 and many other pages.
Brillhart, John; Morton, Patrick. Über Summen von Rudin-Shapiroschen Koeffizienten. (German) Illinois J. Math. 22 (1978), no. 1, 126--148. MR0476686 (57 #16245) - From N. J. A. Sloane, Jun 06 2012
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
Currie, James D.; Rampersad, Narad; Saari, Kalle; Zamboni, Luca Q. Extremal words in morphic subshifts. Discrete Math. 322 (2014), 53--60. MR3164037. See Sect. 8.
Arturas Dubickas, Heights of squares of Littlewood polynomials and infinite series, Ann. Polon. Math. 105 (2012), 145-163. - From N. J. A. Sloane, Dec 16 2012
A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences.
Mendès France, M.; Tenenbaum, G. Dimension des courbes planes, papiers pliés et suites de Rudin-Shapiro. (French) Bull. Soc. Math. France 109 (1981), no. 2, 207--215. MR0623789 (82k:10073).
Joerg Arndt, Fxtbook, section 1.16.5 "The Golay-Rudin-Shapiro sequence", pp.44-45
Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence
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FORMULA
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a(0) = 1, a(2n) = a(n), a(2n+1) = a(n) * (-1)^n.
a(0) = a(1) = 1, a(2n) = a(n), a(2n+1) = a(n)*(1-2*(n AND 1)), where AND is the bitwise AND operator. - Alex Ratushnyak, May 13 2012
Brillhart and Morton (1978) list many properties.
a(n) = (-1)^A014081(n) = (-1)^A020987(n) = 1-2*A020987(n). - M. F. Hasler, Jun 06 2012
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MAPLE
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A020985 := proc(n) option remember; if n = 0 then 1 elif n mod 2 = 0 then A020985(n/2) else (-1)^((n-1)/2 )*A020985( (n-1)/2 ); fi; end;
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MATHEMATICA
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a[0] = 1; a[1] = 1; a[n_?EvenQ] := a[n] = a[n/2]; a[n_?OddQ] := a[n] = (-1)^((n-1)/2)* a[(n-1)/2]; a /@ Range[0, 80] (* Jean-François Alcover, Jul 05 2011 *)
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PROG
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(Haskell)
a020985 n = a020985_list !! n
a020985_list = 1 : 1 : f (tail a020985_list) (-1) where
f (x:xs) w = x : x*w : f xs (0 - w)
-- Reinhard Zumkeller, Jan 02 2012
(PARI) A020985(n)=(-1)^A014081(n) \\ - M. F. Hasler, Jun 06 2012
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CROSSREFS
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Cf. A022155, A005943.
a(n) = (-1)^A014081(n).
Cf. A020987 (0-1 version), A020986 (partial sums), A203531 (run lengths), A033999.
Sequence in context: A108784 * A034947 A097807 A014077 A174351 A181432
Adjacent sequences: A020982 A020983 A020984 * A020986 A020987 A020988
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KEYWORD
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sign,nice,easy
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AUTHOR
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N. J. A. Sloane. Minor edits by N. J. A. Sloane, Jun 06 2012
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STATUS
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approved
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