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A247303 Convolution of A010059(n) with itself. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 28 17:08 EST 2023. Contains 359895 sequences. (Running on oeis4.)