A247303 - OEIS

%I #38 Jun 23 2021 12:05:15

%S 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 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 20,15,10,21,10,10,32,10,12,23,10,19,22,16,16,21,16

%N Convolution of A010059(n) with itself.

%C The parity of this sequence is A228495(n+1).

%C 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 Reinhard Zumkeller, <a href="/A247303/b247303.txt">Table of n, a(n) for n = 0..10000</a>

%H Tanya Khovanova, <a href="http://arxiv.org/abs/1410.2193">There are no coincidences</a>, arXiv:1410.2193 [math.CO], 2014.

%F G.f.: (1/4)*(1/(1 - x) + Product_{k>=0} (1 - x^(2^k)))^2. - _Ilya Gutkovskiy_, Apr 03 2019

%F a(n) = Sum_{k=0..n} (1-A010060(k))*(1-A010060(n-k)), for n>=0. - _G. C. Greubel_, Apr 03 2019

%F From _Jeffrey Shallit_, Jun 22 2021: (Start)

%F a(n) = v mu(x) w, where x is n expressed in base 2, and

%F     v =  [ 1, 0, 0, 0, 0, 0]

%F mu(0) = [[ 1, 0, 0, 0, 0, 0],

%F          [ 0, 0, 1, 0, 0, 0],

%F          [ 0, 0, 0, 0, 1, 0],

%F          [-1, 2,-2, 1, 0, 1],

%F          [-2, 2, 0, 0,-1, 2],

%F          [-1, 2,-3, 0, 1, 2]]

%F mu(1) = [[ 0, 1, 0, 0, 0, 0],

%F          [ 0, 0, 0, 1, 0, 0],

%F          [ 0, 0, 0, 0, 0, 1],

%F          [ 0, 1,-1,-2, 1, 2],

%F          [-2, 2,-2, 2, 0, 1],

%F          [-2, 3,-2,-1, 0, 3]]

%F     w =  [ 1, 0, 0, 2, 0, 2] (End)

%t a59[n_]:= Mod[SeriesCoefficient[(1+Sqrt[(1-3x)/(1+x)])/(2(1+x)), {x, 0, n}], 2];

%t a[n_] := Sum[a59[k] a59[n-k], {k, 0, n}];

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Dec 15 2018 *)

%t Table[Sum[(1-ThueMorse[k])*(1-ThueMorse[n - k]), {k, 0, n}], {n, 0, 80}] (* _G. C. Greubel_, Apr 03 2019 *)

%o (PARI) nh(n)=!(hammingweight(n)%2);

%o a(n) = sum(k=0, n, nh(k)*nh(n-k)); \\ _Michel Marcus_, Sep 12 2014

%o (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 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 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 (Haskell)

%o a247303 n = a247303_list !! n

%o a247303_list = f [head a010059_list] $ tail a010059_list where

%o    f xs (z:zs) = (sum $ zipWith (*) xs (reverse xs)) : f (z : xs) zs

%o -- _Reinhard Zumkeller_, Sep 14 2014

%o (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 Cf. A010059, A108804.

%K nonn,easy

%O 0,4

%A _Tanya Khovanova_, Sep 11 2014

%E More terms from _Michel Marcus_, Sep 12 2014