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A020985 The Rudin-Shapiro or Golay-Rudin-Shapiro sequence. 12
1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1, -1, -1, 1, -1, 1, 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

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]

REFERENCES

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 78 and many other pages.

LINKS

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

J.-P. Allouche, Lecture notes on automatic sequences, Krakow October 2013.

J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences.

Jean-Paul Allouche and Jonathan Sondow, Summation of rational series twisted by strongly B-multiplicative coefficients, arXiv:1408.5770 [math.NT], 2014.

Joerg Arndt, Matters Computational (The Fxtbook), section 1.16.5 "The Golay-Rudin-Shapiro sequence", pp.44-45

John Brillhart, Patrick Morton, 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.

James D. Currie, Narad Rampersad, Kalle Saari, Luca Q. Zamboni, Extremal words in morphic subshifts, arXiv:1301.4972 [math.CO], 2013.

James D. Currie, Narad Rampersad, Kalle Saari, Luca Q. Zamboni, 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.

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

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).

Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence

FORMULA

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

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;

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] (* Jean-François Alcover, Jul 05 2011 *)

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

(PARI) A020985(n)=(-1)^A014081(n)  \\ M. F. Hasler, Jun 06 2012

CROSSREFS

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

KEYWORD

sign,nice,easy

AUTHOR

N. J. A. Sloane. Minor edits by N. J. A. Sloane, Jun 06 2012

STATUS

approved

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Last modified November 28 15:24 EST 2014. Contains 250363 sequences.