A247303 Convolution of A010059(n) with itself.

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

Author

Tanya Khovanova, Sep 11 2014

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

Cf. A010059, A108804.

Programs

Haskell

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

Mathematica

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 *)

PARI

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

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];
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

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

Formula

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)

Extensions

More terms from Michel Marcus, Sep 12 2014