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A247303
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Convolution of A010059(n) with itself.
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2
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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)
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: (1/4)*(1/(1 - x) + Product_{k>=0} (1 - x^(2^k)))^2. - Ilya Gutkovskiy, Apr 03 2019
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)
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MATHEMATICA
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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[Sum[(1-ThueMorse[k])*(1-ThueMorse[n - k]), {k, 0, n}], {n, 0, 80}] (* G. C. Greubel, Apr 03 2019 *)
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PROG
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(PARI) nh(n)=!(hammingweight(n)%2);
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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