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A330033 a(n) = Kronecker(n, 5) * (-1)^floor(n/5).

%I A330033 #21 Jul 25 2025 19:17:16
%S A330033 0,1,-1,-1,1,0,-1,1,1,-1,0,1,-1,-1,1,0,-1,1,1,-1,0,1,-1,-1,1,0,-1,1,1,
%T A330033 -1,0,1,-1,-1,1,0,-1,1,1,-1,0,1,-1,-1,1,0,-1,1,1,-1,0,1,-1,-1,1,0,-1,
%U A330033 1,1,-1,0,1,-1,-1,1,0,-1,1,1,-1,0,1,-1,-1,1,0,-1
%N A330033 a(n) = Kronecker(n, 5) * (-1)^floor(n/5).
%C A330033 This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = y = z = -1.
%H A330033 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%H A330033 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,-1).
%F A330033 Euler transform of length 10 sequence [-1, -1, 0, 0, -1, 0, 0, 0, 0, 1].
%F A330033 G.f.: (x - 2*x^2 + x^3) / (1 - x + x^2 - x^3 + x^4) = x * (1 - x) * (1 - x^2) / (1 + x^5).
%F A330033 a(n) = -a(n+5) = -a(-n) = -(-1)^n*A244895(n) = A080891(n) * A330025(n), |a(n)| = A011558(n) for all n in Z.
%F A330033 a(n) = -A292301(n-1). a(5*n) = 0.
%F A330033 0 = a(n)*a(n-4) - a(n-1)*a(n-3) - a(n-2)*a(n-2) for all n in Z.
%F A330033 0 = a(n)*a(n+5) + a(n+1)*a(n+4) - a(n+2)*a(n+3) for all n in Z.
%e A330033 G.f. = x - x^2 - x^3 + x^4 - x^6 + x^7 + x^8 - x^9 + x^11 - x^12 + ...
%t A330033 a[ n_] := {1, -1, -1, 1, 0}[[Mod[n, 5, 1]]] (-1)^Quotient[n, 5];
%t A330033 a[ n_] := JacobiSymbol[n, 5] (-1)^Quotient[n, 5];
%t A330033 LinearRecurrence[{1,-1,1,-1},{0,1,-1,-1},100] (* _Harvey P. Dale_, Jul 25 2025 *)
%o A330033 (PARI) {a(n) = [0, 1, -1, -1, 1][n%5 + 1] * (-1)^(n\5)};
%o A330033 (PARI) {a(n) = kronecker(n, 5) * (-1)^(n\5)};
%o A330033 (Magma) [KroneckerSymbol(n, 5) * (-1)^Floor(n/5):n in [0..76]]; // _Marius A. Burtea_, Nov 28 2019
%Y A330033 Cf. A011558, A080891, A244895, A292301, A330025.
%K A330033 sign,easy
%O A330033 0,1
%A A330033 _Michael Somos_, Nov 27 2019