A020987 Zero-one version of Golay-Rudin-Shapiro sequence (or word).

0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

This is (1-A020985(n))/2. See A020985, which is the main entry for this sequence, for more information. N. J. A. Sloane, Jun 06 2012

A word that is uniform primitive morphic, but not pure morphic. - N. J. A. Sloane, Jul 14 2018

References

• J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 78.
• Dekking, Michel, Michel Mendes France, and Alf van der Poorten. "Folds." The Mathematical Intelligencer, 4.3 (1982): 130-138 & front cover, and 4:4 (1982): 173-181 (printed in two parts).
• G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
• Lipshitz, Leonard, and A. van der Poorten. "Rational functions, diagonals, automata and arithmetic." In Number Theory, Richard A. Mollin, ed., Walter de Gruyter, Berlin (1990): 339-358.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Jean-Paul Allouche, Schrödinger Operators with Rudin-Shapiro Potentials are not Palindromic, Journal of Mathematical Physics, volume 38, number 4, 1997, pages 1843-1848. And the author's copy. Section IV v_n = a(n) for the Rudin-Shapiro case given there i_0 = 0 and otherwise i_m = m+1 mod 2.
• Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017
• 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, Discrete Math. 322 (2014), 53--60. MR3164037. See Sect. 8.
• Michael Gilleland, Some Self-Similar Integer Sequences
• A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.
• L. Lipshitz and A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic
• H. Niederreiter and M. Vielhaber, Tree complexity and a doubly exponential gap between structured and random sequences, J. Complexity, 12 (1996), 187-198.
• Aayush Rajasekaran, Narad Rampersad, Jeffrey Shallit, Overpals, Underlaps, and Underpals, In: Brlek S., Dolce F., Reutenauer C., Vandomme É. (eds) Combinatorics on Words, WORDS 2017, Lecture Notes in Computer Science, vol 10432.
• Thomas Stoll, On digital blocks of polynomial values and extractions in the Rudin-Shapiro sequence, RAIRO - Theoretical Informatics and Applications (RAIRO: ITA), EDP Sciences, 2016, 50, pp. 93-99. .
• Index entries for characteristic functions

Crossrefs

Cf. A020985.

A014081(n) mod 2. Characteristic function of A022155.

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

a020987 = (`div` 2) . (1 -) . a020985  -- Reinhard Zumkeller, Jun 06 2012

Mathematica

a[n_] := (1/2)*(1-(-1)^Count[Partition[IntegerDigits[n, 2], 2, 1], {1, 1}]); Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Dec 12 2014, after Robert G. Wilson v *)
(1 - RudinShapiro[Range[0, 100]])/2 (* Paolo Xausa, Jan 29 2025 *)

Python

def A020987(n): return (n&(n>>1)).bit_count()&1 # Chai Wah Wu, Feb 11 2023