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A124395
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Expansion of (1-2*x)/(1-2*x+2*x^3).
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3
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1, 0, 0, -2, -4, -8, -12, -16, -16, -8, 16, 64, 144, 256, 384, 480, 448, 128, -704, -2304, -4864, -8320, -12032, -14336, -12032, 0, 28672, 81408, 162816, 268288, 373760, 421888, 307200, -133120, -1110016, -2834432, -5402624, -8585216, -11501568, -12197888
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OFFSET
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0,4
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COMMENTS
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Diagonal sums of number array A124394.
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LINKS
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FORMULA
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a(n) = Sum_{k=0..floor(n/2)} Sum{j=0..k+1} C(k+1,j)*C(n-j+1,2k+1)*(-2)^j.
a(n) = term (2,2) in the 3 X 3 matrix [2,1,0; 0,0,1; -2,0,0]^n. - Alois P. Heinz, Sep 10 2008
a(n) = 2*a(n-1) - 2*a(n-3); a(0)=1, a(1)=0, a(2)=0. - Harvey P. Dale, Dec 21 2013
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MAPLE
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a:= n-> (Matrix([[2, 1, 0], [0, 0, 1], [-2, 0, 0]])^n)[2, 2]: seq (a(n), n=0..35); # Alois P. Heinz, Sep 10 2008
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MATHEMATICA
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CoefficientList[Series[(1-2x)/(1-2x+2x^3), {x, 0, 50}], x] (* or *) LinearRecurrence[{2, 0, -2}, {1, 0, 0}, 50] (* Harvey P. Dale, Dec 21 2013 *)
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PROG
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(PARI) my(x='x+O('x^50)); Vec((1-2*x)/(1-2*x+2*x^3)) \\ G. C. Greubel, Dec 25 2019
(Magma) I:=[1, 0, 0]; [n le 3 select I[n] else 2*Self(n-1) - 2*Self(n-3): n in [1..50]]; // G. C. Greubel, Dec 25 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-2*x)/(1-2*x+2*x^3) ).list()
(GAP) a:=[1, 0, 0];; for n in [3..50] do a[n]:=2*a[n-1]-2*a[n-3]; od; a; # G. C. Greubel, Dec 25 2019
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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