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%I #36 Feb 12 2023 03:12:41
%S 1,0,1,2,3,2,1,0,1,0,1,2,1,2,3,4,5,4,5,6,7,6,5,4,3,4,3,2,3,2,1,0,1,0,
%T 1,2,3,2,1,0,1,0,1,2,1,2,3,4,3,4,3,2,1,2,3,4,5,4,5,6,5,6,7,8,9,8,9,10,
%U 11,10,9,8,9,8,9,10,9,10,11
%N a(n) = Sum_{k=0..n} (-1)^k*A020985(k).
%H Reinhard Zumkeller, <a href="/A020990/b020990.txt">Table of n, a(n) for n = 0..10000</a>
%H John Brillhart and Patrick Morton, <a href="http://projecteuclid.org/euclid.ijm/1256048841">Über Summen von Rudin-Shapiroschen Koeffizienten</a>, (German) Illinois J. Math. 22 (1978), no. 1, 126--148. MR0476686 (57 #16245). - _N. J. A. Sloane_, Jun 06 2012
%H J. Brillhart and P. Morton, <a href="http://www.maa.org/programs/maa-awards/writing-awards/a-case-study-in-mathematical-research-the-golay-rudin-shapiro-sequence">A case study in mathematical research: the Golay-Rudin-Shapiro sequence</a>, Amer. Math. Monthly, 103 (1996) 854-869.
%H Narad Rampersad and Jeffrey Shallit, <a href="https://arxiv.org/abs/2302.00405">Rudin-Shapiro Sums Via Automata Theory and Logic</a>, arXiv:2302.00405 [math.NT], February 1 2023.
%H <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F Brillhart and Morton (1978) list many properties.
%o (Haskell)
%o a020990 n = a020990_list !! n
%o a020990_list = scanl1 (+) $ zipWith (*) a033999_list a020985_list
%o -- _Reinhard Zumkeller_, Jun 06 2012
%o (PARI) a(n) = sum(k=0, n, (-1)^(k+hammingweight(bitand(k, k>>1)))); \\ _Michel Marcus_, Oct 07 2017
%o (Python)
%o def A020990(n): return sum(-1 if ((m&(m>>1)).bit_count()^m)&1 else 1 for m in range(n+1)) # _Chai Wah Wu_, Feb 11 2023
%Y Cf. A033999, A020986.
%K nonn
%O 0,4
%A _N. J. A. Sloane_
%E Edited by _N. J. A. Sloane_, Jun 06 2012
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