A327767 Period 2: repeat [1, -2].

1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2

Offset: 1

Author

Michael Somos, Sep 24 2019

Keywords

• sign
• mult
• easy

Examples

			G.f. = x - 2*x^2 + x^3 - 2*x^4 + x^5 - 2*x^6 + x^7 - 2*x^8 + ...
		

Links

• Michael Somos, Rational Function Multiplicative Coefficients.
• Index entries for linear recurrences with constant coefficients, signature (0,1).

Crossrefs

Cf. A000034, A040001, A134451, A168361, A280193.

Programs

Magma

&cat [[1, -2]^^50]; // Vincenzo Librandi, Feb 29 2020

Mathematica

a[ n_] := If[ n < 1, 0, -2 + 3 Mod[n, 2]];
a[ n_] := Which[ n < 1, 0, OddQ[n], 1, True, -2];
a[ n_] := SeriesCoefficient[ (x - 2*x^2) / (1 - x^2), {x, 0, n}];
PadRight[{}, 100, {1, -2}] (* Vincenzo Librandi, Feb 29 2020 *)

PARI

{a(n) = if( n<1, 0, -(1 + 3*(-1)^n)/2)};

PARI

{a(n) = if( n<1, 0, -2 + 3*(n%2))};

PARI

{a(n) = if( n<1, 0, [-2, 1][n%2 + 1])};

PARI

{a(n) = if( n<0, 0, polcoeff( (x - 2*x^2) / (1 - x^2) + x * O(x^n), n))};

Formula

G.f.: x * (1 - 2*x) / (1 - x^2) = x / (1 + 2*x / (1 - 3*x / (2 - x))).

E.g.f.: (exp(x) - 1)*(3/exp(x) - 1)/2.

a(n) is multiplicative with a(2^e) = -2 if e>0, a(p^e) = 1 otherwise.

Moebius transform is length 2 sequence [1, -3].

a(n) = -(1 + 3*(-1)^n)/2 if n>=1.

a(2*n) = -2, a(2*n + 1) = 1, a(0) = 0.

a(n) = -(-1)^n * A134451(n) for all n in Z.

a(n) = a(n+2) = -(-1)^n * A000034(n-1) = -A168361(n+1) for n>=1.

Dirichlet g.f.: zeta(s)*(1-3/2^s). - Amiram Eldar, Jan 03 2023