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A020985
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The Golay-Rudin-Shapiro sequence.
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8
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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; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| a(A022155(n)) = -1; a(A203463(n)) = 1. [Reinhard Zumkeller, Jan 02 2012]
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REFERENCES
| J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 78.
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
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
| Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Joerg Arndt, Fxtbook
J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences.
Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence
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FORMULA
| a_0=1, a_2n = a_n, a_2n+1 = (-1)^n *a_n.
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MAPLE
| 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
| 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] (* From Jean-François Alcover, Jul 05 2011 *)
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PROG
| (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
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CROSSREFS
| Cf. A022155.
(-1)^A014081(n).
Cf. 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
| sign,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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