A230074 Period 4: repeat [1, -2, 1, 0].

1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2

Offset: 1

Author

Wolfdieter Lang, Oct 21 2013

Keywords

• sign
• easy
• mult

Comments

The o.g.f. for this sequence is obtained from the o.g.f.'s of the bisection of the sequence including a(0) = 0.

For the cos product formula below use Product_{k=1..n-1} 2*cos(2*k*Pi/n) = 1 if n is odd, and Product_{k=1..n-1} 2*cos(2*k*Pi/n) = -(1-(-1)^(n/2)) if n is even (see Gradstein-Rhyzik, p.62, 1.393 1., with x=0).

Also, the arithmetic function uhat(n,4,4) as defined in A291041. - Robert Price, Aug 25 2017

References

• I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.

Links

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

Crossrefs

Cf. A291041.

Programs

Magma

&cat [[1, -2, 1, 0]^^30]; // Wesley Ivan Hurt, Jul 09 2016

Maple

A230074:=n->sqrt(n^2 mod 8)*(-1)^(n+1); seq(A230074(n), n=1..100); # Wesley Ivan Hurt, Jan 01 2014

Mathematica

Table[Sqrt[Mod[n^2, 8]](-1)^(n+1), {n, 100}] (* Wesley Ivan Hurt, Jan 01 2014 *)
PadRight[{},120,{1,-2,1,0}] (* Harvey P. Dale, Apr 17 2014 *)

PARI

a(n)=([0,1,0; 0,0,1; -1,-1,-1]^(n-1)*[1;-2;1])[1,1] \\ Charles R Greathouse IV, Jul 17 2016

Formula

a(n) = 1 if n is odd, and a(n) = -(1-(-1)^(n/2)) if n is even.

a(n+4*k) = a(n), n = 1, 2, 3, 4, k >= 1.

G.f.: -2*x/(1-x^4) + 1/(1-x^2) = (1-x)/((1+x)*(1+x^2)).

a(n) = Product_{k=1..n-1} 2*cos(2*k*Pi/n).

a(n) = sqrt(n^2 mod 8)*(-1)^(n+1). - Wesley Ivan Hurt, Jan 01 2014

From Wesley Ivan Hurt, Jun 22 2015: (Start)

a(n) + a(n-1) + a(n-2) + a(n-3) = 0, n>3.

a(n) = (1+(-1)^n)*(-1)^(n/2)/2-(-1)^n. (End)

From Wesley Ivan Hurt, Jul 09 2016: (Start)

a(n) = a(n-4) for n>4.

a(n) = cos(n*Pi/2) - (-1)^n. (End)

E.g.f.: cos(x) - exp(-x). - Ilya Gutkovskiy, Jul 09 2016

From Werner Schulte, Aug 29 2020: (Start)

Multiplicative with a(2^e) = (-2)^e if e<2 and 0 if e>1, and a(p^e) = 1 for prime p > 2.

Dirichlet g.f.: zeta(s) * (1-2^(-s)) * (1-2^(1-s)).

Dirichlet inverse b(n) is multiplicative with b(2^e) = 2^e and, for prime p>2, b(p^e) = (-1)^e if e<2 and 0 if e>1. (End)