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
Offset: 0
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
Programs
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Haskell
a020986 n = a020986_list !! n a020986_list = scanl1 (+) a020985_list -- Reinhard Zumkeller, Jan 02 2012
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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 *) -
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
Formula
Brillhart and Morton (1978) list many properties.
Extensions
Minor edits by N. J. A. Sloane, Jun 06 2012