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A020985
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The Rudin-Shapiro or Golay-Rudin-Shapiro sequence (coefficients of the Shapiro polynomials).
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42
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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
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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.
The Shapiro polynomials are defined by P_0 = Q_0 = 1; for n>=0, P_{n+1} = P_n + x^(2^n)*Q_n, Q_{n+1} = P_n - x^(2^n)*Q_n. Then P_n = Sum_{m=0..2^n-1} a(m)*x^m, where the a(m) (the present sequence) do not depend on n. - N. J. A. Sloane, Aug 12 2016
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
a(n) = 1 if and only if the numbers of 1's and runs of 1's in binary representation of n have the same parity: A010060(n) = A268411(n); otherwise, when A010060(n) = 1 - A268411(n), a(n) = -1. - Vladimir Shevelev, Feb 10 2016. Typo corrected and comment edited by Antti Karttunen, Jul 11 2017
A word that is uniform primitive morphic, but not pure morphic. - N. J. A. Sloane, Jul 14 2018
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 78 and many other pages.
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LINKS
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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, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807, Nov 29 2017
Jean-Paul Allouche and Jonathan Sondow, Summation of rational series twisted by strongly B-multiplicative coefficients, arXiv:1408.5770 [math.NT], 2014; Electron. J. Combin., 22 #1 (2015) P1.59; see pp.9-10.
Joerg Arndt, Matters Computational (The Fxtbook), section 1.16.5 "The Golay-Rudin-Shapiro sequence", pp.44-45
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
J. Brillhart and L. Carlitz, Note on the Shapiro polynomials, Proc. Amer. Math. Soc. 25 (1970), 114-118.
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
John Brillhart and Patrick 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.
D. H. Lehmer, and Emma Lehmer, Picturesque exponential sums. II, Journal für die reine und angewandte Mathematik, 318 (1980), 1-19.
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).
Luke Schaeffer, Jeffrey Shallit, Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences, Electronic Journal of Combinatorics 23(1) (2016), #P1.25.
Harold S. Shapiro, Extremal problems for polynomials and power series, Ph.D. Diss. Massachusetts Institute of Technology, 1952.
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.
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. [Brillhart and Carlitz, in proof of theorem 4]
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
Sum(n >= 1, a(n-1)(8n^2+4n+1)/(2n(2n+1)(4n+1)) = 1; see Allouche and Sondow, 2015. - Jean-Paul Allouche and Jonathan Sondow, Mar 21 2015
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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 *)
a[n_] := 1 - 2 Mod[Length[FixedPointList[BitAnd[#, # - 1] &, BitAnd[n, Quotient[n, 2]]]], 2] (* Jan Mangaldan, Jul 23 2015 *)
Array[RudinShapiro, 81, 0] (* JungHwan Min, Dec 22 2016 *)
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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
(Python)
def a014081(n): return sum([((n>>i)&3==3) for i in range(len(bin(n)[2:]) - 1)])
def a(n): return (-1)**a014081(n) # Indranil Ghosh, Jun 03 2017
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CROSSREFS
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Cf. A022155, A005943 (factor complexity), A014081.
Cf. A020987 (0-1 version), A020986 (partial sums), A203531 (run lengths), A033999.
Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.
Sequence in context: A010555 A108784 A244513 * A034947 A097807 A014077
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
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STATUS
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approved
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