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%I #18 Nov 16 2023 05:35:23
%S 1,0,1,-1,0,-1,-1,0,-1,1,0,1,1,0,1,-1,0,-1,-1,0,-1,1,0,1,1,0,1,-1,0,
%T -1,-1,0,-1,1,0,1,1,0,1,-1,0,-1,-1,0,-1,1,0,1,1,0,1,-1,0,-1,-1,0,-1,1,
%U 0,1,1,0,1,-1,0,-1,-1,0,-1,1,0,1,1,0,1,-1,0,-1
%N Kronecker symbol(-6 / 2*n + 7).
%H G. C. Greubel, <a href="/A260192/b260192.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,-1).
%F Euler transform of length 12 sequence [ 0, 1, -1, -1, 0, -1, 0, 0, 0, 0, 0, 1].
%F G.f.: (1 - x^3) / (1 - x^2 + x^4).
%F a(n) = -(-1)^n * a(n+3) = -a(n+6) = a(-1-n) = a(n+2) - a(n+4) for all n in Z.
%F a(n) = = (-1)^n * A260190(n) = A117441(n+1) = A109017(2*n + 7).
%F a(2*n) = A010892(n). a(2*n + 1) = A128834(n). a(3*n + 1) = 0. a(3*n) = a(3*n + 2) = A087960(n).
%e G.f. = 1 + x^2 - x^3 - x^5 - x^6 - x^8 + x^9 + x^11 + x^12 + x^14 - x^15 + ...
%t a[ n_] := KroneckerSymbol[ -6, 2 n + 7];
%t LinearRecurrence[{0, 1, 0, -1}, {1, 0, 1, -1}, 50] (* _G. C. Greubel_, Jan 15 2018 *)
%t CoefficientList[Series[(1-x^3)/(1-x^2+x^4),{x,0,100}],x] (* _Harvey P. Dale_, Jun 30 2021 *)
%o (PARI) {a(n) = kronecker( -6, 2*n + 7)};
%o (PARI) {a(n) = (-1)^(n\6 + n) * [1, 0, 1][n%3 + 1]};
%o (PARI) {a(n) = if( n<0, n=-1-n); polcoeff( (1 - x^3) / (1 - x^2 + x^4) + x * O(x^n), n)};
%o (Magma) I:=[1,0,1,-1]; [n le 4 select I[n] else Self(n-2) - Self(n-4): n in [1..30]]; // _G. C. Greubel_, Jan 15 2018
%Y Cf. A087960, A109017, A128834, A260190.
%Y Essentially the same as A117441.
%K sign,easy
%O 0
%A _Michael Somos_, Jul 18 2015
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