A108804 -id:A108804 - cp's OEIS frontend

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

Offset: 0

Tanya Khovanova, Sep 11 2014

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

Alternatively, the number of ways to write n = x+y, where x, y are evil numbers (members of A001969). - Jeffrey Shallit, Jun 22 2021

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Tanya Khovanova, There are no coincidences, arXiv:1410.2193 [math.CO], 2014.

Cf. A010059, A108804.

a247303 n = a247303_list !! n
a247303_list = f [head a010059_list] $ tail a010059_list where
   f xs (z:zs) = (sum $ zipWith (*) xs (reverse xs)) : f (z : xs) zs
-- Reinhard Zumkeller, Sep 14 2014

a59[n_]:= Mod[SeriesCoefficient[(1+Sqrt[(1-3x)/(1+x)])/(2(1+x)), {x, 0, n}], 2];
a[n_] := Sum[a59[k] a59[n-k], {k, 0, n}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 15 2018 *)
Table[Sum[(1-ThueMorse[k])*(1-ThueMorse[n - k]), {k, 0, n}], {n, 0, 80}] (* G. C. Greubel, Apr 03 2019 *)

nh(n)=!(hammingweight(n)%2);
a(n) = sum(k=0, n, nh(k)*nh(n-k)); \\ Michel Marcus, Sep 12 2014

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];
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];
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

[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

G.f.: (1/4)*(1/(1 - x) + Product_{k>=0} (1 - x^(2^k)))^2. - Ilya Gutkovskiy, Apr 03 2019
a(n) = Sum_{k=0..n} (1-A010060(k))*(1-A010060(n-k)), for n>=0. - G. C. Greubel, Apr 03 2019
From Jeffrey Shallit, Jun 22 2021: (Start)
a(n) = v mu(x) w, where x is n expressed in base 2, and
    v =  [ 1, 0, 0, 0, 0, 0]
mu(0) = [[ 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]]
mu(1) = [[ 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]]
    w =  [ 1, 0, 0, 2, 0, 2] (End)

More terms from Michel Marcus, Sep 12 2014

cp's OEIS Frontend

A108805 A108804 read mod 4.

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A247303 Convolution of A010059(n) with itself.

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