A203463 -id:A203463 - cp's OEIS frontend

A020985 The Rudin-Shapiro or Golay-Rudin-Shapiro sequence (coefficients of the Shapiro polynomials).

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

Offset: 0

N. J. A. Sloane

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
Named after the Austrian-American mathematician Walter Rudin (1921-2010), the mathematician Harold S. Shapiro (1928-2021) and the Swiss mathematician and physicist Marcel Jules Edouard Golay (1902-1989). - Amiram Eldar, Jun 13 2021

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 78 and many other pages.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche, Lecture notes on automatic sequences, Krakow October 2013.
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017
Jean-Paul Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, Vol. 3, Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11.
Jean-Paul Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. [Local copy]
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.
John Brillhart and L. Carlitz, Note on the Shapiro polynomials, Proc. Amer. Math. Soc., Vol. 25 (1970), pp. 114-118.
John Brillhart and Patrick Morton, Über Summen von Rudin-Shapiroschen Koeffizienten, (German) Illinois J. Math., Vol. 22, No. 1 (1978), pp. 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, Vol. 103 (1996) pp. 854-869.
James D. Currie, Narad Rampersad, Kalle Saari, and Luca Q. Zamboni, Extremal words in morphic subshifts, arXiv:1301.4972 [math.CO], 2013.
James D. Currie, Narad Rampersad, Kalle Saari, and Luca Q. Zamboni, Extremal words in morphic subshifts, Discrete Math., Vol. 322 (2014), pp. 53-60. MR3164037. See Sect. 8.
Michel Dekking, Michel Mendes France and Alf van der Poorten, Folds, The Mathematical Intelligencer, Vol. 4, No. 3 (1982), pp. 130-138.
Michel Dekking, Michel Mendes France and Alf van der Poorten, Folds II. Symmetry disturbed, The Mathematical Intelligencer, Vol. 4, No. 4 (1982), pp. 173-181.
Arturas Dubickas, Heights of squares of Littlewood polynomials and infinite series, Ann. Polon. Math., Vol. 105 (2012), pp. 145-163. - From _N. J. A. Sloane_, Dec 16 2012
Russell Jay Hendel, A Family of Sequences Generalizing the Thue Morse and Rudin Shapiro Sequences, arXiv:2505.20547 [cs.FL], 2025. See p. 2.
Albertus Hof, Oliver Knill and Barry Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys., Vol. 174, No. 1 (1995), pp. 149-159.
Philip Lafrance, Narad Rampersad and 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, Vol. 318 (1980), pp. 1-19.
Michel Mendès France and Gérald Tenenbaum, Dimension des courbes planes, papiers pliés et suites de Rudin-Shapiro. (French) Bull. Soc. Math. France, Vol. 109, No. 2 (1981), pp. 207-215. MR0623789 (82k:10073).
Luke Schaeffer and Jeffrey Shallit, Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences, Electronic Journal of Combinatorics, Vol. 23, No. 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.

Cf. A022155, A005943 (factor complexity), A014081.
Cf. A020987 (0-1 version), A020986 (partial sums), A203531 (run lengths), A033999, A380667 (first differences).
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.

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

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;

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 *)

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

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

def A020985(n): return -1 if (n&(n>>1)).bit_count()&1 else 1 # Chai Wah Wu, Feb 11 2023

a(0) = a(1) = 1; thereafter, 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)*(8*n^2+4*n+1)/(2*n*(2*n+1)*(4*n+1)) = 1; see Allouche and Sondow, 2015. - Jean-Paul Allouche and Jonathan Sondow, Mar 21 2015

A022155 Values of n at which Golay-Rudin-Shapiro sequence A020985 is negative.

Original entry on oeis.org

3, 6, 11, 12, 13, 15, 19, 22, 24, 25, 26, 30, 35, 38, 43, 44, 45, 47, 48, 49, 50, 52, 53, 55, 59, 60, 61, 63, 67, 70, 75, 76, 77, 79, 83, 86, 88, 89, 90, 94, 96, 97, 98, 100, 101, 103, 104, 105, 106, 110, 115, 118, 120, 121, 122, 126, 131, 134, 139, 140

Offset: 1

Robert G. Wilson v

nonn

A020985(a(n)) = -1.

Or numbers n for which numbers of 1's and runs of 1's in binary representation have distinct parities: A010060(n) = 1 - A268411(n). - Vladimir Shevelev, Feb 10 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.

Cf. A203463 (complement), A020985.

import Data.List (elemIndices)
a022155 n = a022155_list !! (n-1)
a022155_list = elemIndices (- 1) a020985_list
-- Reinhard Zumkeller, Jan 02 2012

Position[Array[RudinShapiro, 200, 0], ?Negative] - 1 // Flatten (* _Jean-François Alcover, Dec 04 2018 *)

from itertools import count, islice
def A022155_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:(n&(n>>1)).bit_count()&1,count(max(startvalue,0)))
A022155_list = list(islice(A022155_gen(),30)) # Chai Wah Wu, Feb 11 2023

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A020985 The Rudin-Shapiro or Golay-Rudin-Shapiro sequence (coefficients of the Shapiro polynomials).

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A022155 Values of n at which Golay-Rudin-Shapiro sequence A020985 is negative.

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