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

Author

N. J. A. Sloane

Keywords

• sign
• nice
• easy

Comments

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

References

• Jean-Paul Allouche and Jeffrey 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
• 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.

Crossrefs

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.

Programs

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

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 *)
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 *)

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

Python

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

Formula

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