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

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, -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, 1, 1, -1, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1, 0, 1, -1, -1, 1, 0, -1

Offset: 0

Author

Michael Somos, Nov 27 2019

Keywords

• sign
• easy

Comments

This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = y = z = -1.

Examples

			G.f. = x - x^2 - x^3 + x^4 - x^6 + x^7 + x^8 - x^9 + x^11 - x^12 + ...
		

Links

• Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
• Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1).

Crossrefs

Cf. A011558, A080891, A244895, A292301, A330025.

Programs

Magma

[KroneckerSymbol(n, 5) * (-1)^Floor(n/5):n in [0..76]]; // Marius A. Burtea, Nov 28 2019

Mathematica

a[ n_] := {1, -1, -1, 1, 0}[[Mod[n, 5, 1]]] (-1)^Quotient[n, 5];
a[ n_] := JacobiSymbol[n, 5] (-1)^Quotient[n, 5];
LinearRecurrence[{1,-1,1,-1},{0,1,-1,-1},100] (* Harvey P. Dale, Jul 25 2025 *)

PARI

{a(n) = [0, 1, -1, -1, 1][n%5 + 1] * (-1)^(n\5)};

PARI

{a(n) = kronecker(n, 5) * (-1)^(n\5)};

Formula

Euler transform of length 10 sequence [-1, -1, 0, 0, -1, 0, 0, 0, 0, 1].

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).

a(n) = -a(n+5) = -a(-n) = -(-1)^n*A244895(n) = A080891(n) * A330025(n), |a(n)| = A011558(n) for all n in Z.

a(n) = -A292301(n-1). a(5*n) = 0.

0 = a(n)*a(n-4) - a(n-1)*a(n-3) - a(n-2)*a(n-2) for all n in Z.

0 = a(n)*a(n+5) + a(n+1)*a(n+4) - a(n+2)*a(n+3) for all n in Z.