A020986 - OEIS

A020986

a(n) = n-th partial sum of Golay-Rudin-Shapiro sequence A020985.

1, 2, 3, 2, 3, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 4, 5, 6, 7, 6, 7, 8, 7, 8, 7, 6, 5, 6, 7, 8, 7, 8, 9, 10, 11, 10, 11, 12, 11, 12, 13, 14, 15, 14, 13, 12, 13, 12, 11, 10, 9, 10, 9, 8, 9, 8, 9, 10, 11, 10, 9, 8, 9, 8, 9, 10, 11, 10, 11, 12, 11, 12, 13, 14, 15, 14, 13, 12, 13, 12, 13, 14, 15, 14, 15, 16

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

John Brillhart and 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.

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

Narad Rampersad and Jeffrey Shallit, Rudin-Shapiro Sums Via Automata Theory and Logic, arXiv:2302.00405 [math.NT], February 1 2023.

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

Index entries for sequences related to coordinates of 2D curves

FORMULA

Brillhart and Morton (1978) list many properties.

MATHEMATICA

a[n_] := 1 - 2 Mod[Length[FixedPointList[BitAnd[#, # - 1] &, BitAnd[n, Quotient[n, 2]]]], 2]; Accumulate@ Table[a@ n, {n, 0, 85}] (* Michael De Vlieger, Nov 30 2015, after Jan Mangaldan at A020985 *)

Table[RudinShapiro[n], {n, 0, 100}] // Accumulate (* Jean-François Alcover, Jun 30 2022 *)

PROG

(Haskell)

a020986 n = a020986_list !! n

a020986_list = scanl1 (+) a020985_list

-- Reinhard Zumkeller, Jan 02 2012

(Python)

def A020986(n): return sum(-1 if (m&(m>>1)).bit_count()&1 else 1 for m in range(n+1)) # Chai Wah Wu, Feb 11 2023

CROSSREFS

Cf. A020985, A020990.

Sequence in context: A363678 A194960 A111439 * A326820 A095161 A072106

Adjacent sequences: A020983 A020984 A020985 * A020987 A020988 A020989