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A247303 Convolution of A010059(n) with itself.

%I A247303 #38 Jun 23 2021 12:05:15
%S A247303 1,0,0,2,0,2,3,0,2,4,3,2,5,2,2,8,2,4,7,2,7,6,4,8,7,4,6,10,4,10,11,0,
%T A247303 10,12,7,10,11,6,8,16,9,8,12,10,10,14,13,8,15,12,10,18,10,14,17,8,14,
%U A247303 20,15,10,21,10,10,32,10,12,23,10,19,22,16,16,21,16
%N A247303 Convolution of A010059(n) with itself.
%C A247303 The parity of this sequence is A228495(n+1).
%C A247303 Alternatively, the number of ways to write n = x+y, where x, y are evil numbers (members of A001969). - _Jeffrey Shallit_, Jun 22 2021
%H A247303 Reinhard Zumkeller, <a href="/A247303/b247303.txt">Table of n, a(n) for n = 0..10000</a>
%H A247303 Tanya Khovanova, <a href="http://arxiv.org/abs/1410.2193">There are no coincidences</a>, arXiv:1410.2193 [math.CO], 2014.
%F A247303 G.f.: (1/4)*(1/(1 - x) + Product_{k>=0} (1 - x^(2^k)))^2. - _Ilya Gutkovskiy_, Apr 03 2019
%F A247303 a(n) = Sum_{k=0..n} (1-A010060(k))*(1-A010060(n-k)), for n>=0. - _G. C. Greubel_, Apr 03 2019
%F A247303 From _Jeffrey Shallit_, Jun 22 2021: (Start)
%F A247303 a(n) = v mu(x) w, where x is n expressed in base 2, and
%F A247303     v =  [ 1, 0, 0, 0, 0, 0]
%F A247303 mu(0) = [[ 1, 0, 0, 0, 0, 0],
%F A247303          [ 0, 0, 1, 0, 0, 0],
%F A247303          [ 0, 0, 0, 0, 1, 0],
%F A247303          [-1, 2,-2, 1, 0, 1],
%F A247303          [-2, 2, 0, 0,-1, 2],
%F A247303          [-1, 2,-3, 0, 1, 2]]
%F A247303 mu(1) = [[ 0, 1, 0, 0, 0, 0],
%F A247303          [ 0, 0, 0, 1, 0, 0],
%F A247303          [ 0, 0, 0, 0, 0, 1],
%F A247303          [ 0, 1,-1,-2, 1, 2],
%F A247303          [-2, 2,-2, 2, 0, 1],
%F A247303          [-2, 3,-2,-1, 0, 3]]
%F A247303     w =  [ 1, 0, 0, 2, 0, 2] (End)
%t A247303 a59[n_]:= Mod[SeriesCoefficient[(1+Sqrt[(1-3x)/(1+x)])/(2(1+x)), {x, 0, n}], 2];
%t A247303 a[n_] := Sum[a59[k] a59[n-k], {k, 0, n}];
%t A247303 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Dec 15 2018 *)
%t A247303 Table[Sum[(1-ThueMorse[k])*(1-ThueMorse[n - k]), {k, 0, n}], {n, 0, 80}] (* _G. C. Greubel_, Apr 03 2019 *)
%o A247303 (PARI) nh(n)=!(hammingweight(n)%2);
%o A247303 a(n) = sum(k=0, n, nh(k)*nh(n-k)); \\ _Michel Marcus_, Sep 12 2014
%o A247303 (PARI) m0 = [1,0,0,0,0,0; 0,0,1,0,0,0; 0,0,0,0,1,0; -1,2,-2,1,0,1; -2,2,0,0,-1,2; -1,2,-3,0,1,2];
%o A247303 m1 = [0,1,0,0,0,0; 0,0,0,1,0,0; 0,0,0,0,0,1; 0,1,-1,-2,1,2; -2,2,-2,2,0,1; -2,3,-2,-1,0,3];
%o A247303 a(n)=my(t=[1,0,0,0,0,0]); forstep(i=exponent(n),0,-1, t*=if(bittest(n,i),m1,m0)); t*[1,0,0,2,0,2]~; \\ Following Shallit; for more efficiency, calculate by bytes instead of bits. _Charles R Greathouse IV_, Jun 23 2021
%o A247303 (Haskell)
%o A247303 a247303 n = a247303_list !! n
%o A247303 a247303_list = f [head a010059_list] $ tail a010059_list where
%o A247303    f xs (z:zs) = (sum $ zipWith (*) xs (reverse xs)) : f (z : xs) zs
%o A247303 -- _Reinhard Zumkeller_, Sep 14 2014
%o A247303 (Sage) [sum((1-sloane.A010060(k))*(1-sloane.A010060(n-k)) for k in (0..n)) for n in (0..80)] # _G. C. Greubel_, Apr 03 2019
%Y A247303 Cf. A010059, A108804.
%K A247303 nonn,easy
%O A247303 0,4
%A A247303 _Tanya Khovanova_, Sep 11 2014
%E A247303 More terms from _Michel Marcus_, Sep 12 2014