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

0,4

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.

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)

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A010059, A108804.

Sequence in context: A141099 A127710 A137510 * A067871 A198632 A060155

Adjacent sequences: A247300 A247301 A247302 * A247304 A247305 A247306

Keyword

nonn,easy

Author

Tanya Khovanova, Sep 11 2014

Extensions

More terms from Michel Marcus, Sep 12 2014

Status

approved