A234044 Period 7: repeat [2, -2, 1, 0, 0, 1, -2].

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

Offset: 0

Author

Wolfdieter Lang, Feb 27 2014

Keywords

• sign
• easy

Comments

This is a member of the six sequences which appear for the instance N=7 of the general formula 2*exp(2*Pi*n*I/N) = R(n, x^2-2) + x*S(n-1, x^2-2)*s(N)*I, for n >= 0, with I = sqrt(-1), s(N) = sqrt(2-x)*sqrt(2+x), x = rho(N) := 2*cos(Pi/N) and R and S are the monic Chebyshev polynomials whose coefficient tables are given in A127672 and A049310. If powers x^k with k >= delta(N) = A055034(N) enter in R or x*S then C(N, x), the minimal polynomial of x = rho(N) (see A187360) is used for a reduction. If delta(N) = 2 it may happen that sqrt(2+x) or sqrt(2-x) is an integer in the number field Q(rho(N)). See the N=5 case comment on A164116.

For N=7 with delta(7) = 3, and C(7, x) = x^3 - x^2 - 2*x + 1 the final result becomes 2*exp(2*Pi*n*I/7) = (a(n) + b(n)*x + c(n)*x^2) + (A(n) + B(n)*x + C(n)*x^2)*s(7)*I, with x = rho(7) = 2*cos(Pi/7), a(n) the present sequence, b(n) = A234045(n), c(n) = A234046(n), A(n) = A238468(n), B(n) = A238469(n) and C(n) = A238470(n). The a, b, c and A, B, C brackets are integers in Q(rho(7)).

Examples

			n = 4: 2*exp(8*Pi*I/7) = (2-16*x^2+20*x^4-8*x^6+x^8) + (4*x+10*x^3-6*x^5+x^7)*s(7)*I, reduced with C(7, x) = x^3 - x^2 - 2*x + 1 = 0 this becomes = (-x) + (-1)*s(7)*I with x= 2*cos(Pi/7) and s(7) = 2*sin(Pi/7).The power basis coefficients are thus (a(4), b(4), c(4)) = (0, -1, 0) and (A(4), B(4), C(4)) = (-1, 0, 0).
		

Links

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

Crossrefs

Cf. A234045, A234046, A238468, A238469, A238470, A099837 (N=3), A056594 (N=4), A164116 (N=5), A057079 (N=6).

Programs

Magma

&cat [[2, -2, 1, 0, 0, 1, -2]^^20]; // Wesley Ivan Hurt, Jul 16 2016

Maple

seq(op([2, -2, 1, 0, 0, 1, -2]), n=0..20); # Wesley Ivan Hurt, Jul 16 2016

Mathematica

PadRight[{}, 100, {2, -2, 1, 0, 0, 1, -2}] (* Wesley Ivan Hurt, Jul 16 2016 *)

PARI

a(n)=[2, -2, 1, 0, 0, 1, -2][n%7+1] \\ Charles R Greathouse IV, Jul 17 2016

Formula

G.f.: (2 - 2*x + x^2 + x^5 - 2*x^6)/(1 - x^7).

a(n+7) = a(n) for n>=0, with a(0) = -a(1) = -a(6) = 2, a(3) = a(4) =0 and a(2) = a(5) = 1.

From Wesley Ivan Hurt, Jul 16 2016: (Start)

a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) = 0 for n>5.

a(n) = (1/7) * Sum_{k=1..6} 2*cos((2k)*n*Pi/7) - 2*cos((2k)*(1+n)*Pi/7) + cos((2k)*(2+n)*Pi/7) + cos((2k)*(5+n)*Pi/7) - 2*cos((2k)*(6+n)*Pi/7).

a(n) = 2 + 4*floor(n/7) - 3*floor((1+n)/7) + floor((2+n)/7) - floor((4+n)/7) + 3*floor((5+n)/7) - 4*floor((6+n)/7). (End)