A280569 a(n) = (-1)^n * 2 if n = 5*k and n!=0, otherwise a(n) = (-1)^n.

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

Offset

0,6

Links

Table of n, a(n) for n=0..74.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,-1).

Formula

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

a(n) = -b(n) where b() is multiplicative with b(2^e) = -1 if e>0, b(5^e) = 2 if e>0, b(p^e) = 1 otherwise.

G.f.: (1 - x + x^2) * (1 - x^3) / (1 + x^5).

G.f.: 1 - x / (1 + x) - x^5 / (1 + x^5).

a(n) = a(-n) for all n in Z.

a(5*n) = A280560(n) for all n in Z.

Example

G.f. = 1 - x + x^2 - x^3 + x^4 - 2*x^5 + x^6 - x^7 + x^8 - x^9 + 2*x^10 + ...

Mathematica

a[ n_] := (-1)^n If[ n != 0 && Divisible[n, 5], 2, 1];
LinearRecurrence[{0, 0, 0, 0, -1}, {1, -1, 1, -1, 1, -2}, 120] (* or *) PadRight[ {1}, 120, {2, -1, 1, -1, 1, -2, 1, -1, 1, -1}] (* Harvey P. Dale, Jul 18 2021 *)

Prog

(PARI) {a(n) = (-1)^n * if(n && n%5==0, 2, 1)};
(PARI) {a(n) = n=abs(n); polcoeff( (1 - x + x^2) * (1 - x^3) / (1 + x^5) + x * O(x^n), n)};
(Magma) m:=75; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 - x+x^2)*(1-x^3)/(1+x^5))); // G. C. Greubel, Jul 29 2018

Crossrefs

Cf. A280560.

Sequence in context: A295632 A139549 A216915 * A140345 A177706 A130782

Adjacent sequences: A280566 A280567 A280568 * A280570 A280571 A280572

Keyword

sign

Author

Michael Somos, Jan 05 2017

Status

approved