A234044 - OEIS

%I #17 Sep 08 2022 08:46:06

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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)).

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,-1,-1,-1,-1).

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

%F 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.

%F From _Wesley Ivan Hurt_, Jul 16 2016: (Start)

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

%F 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).

%F 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)

%e 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).

%p seq(op([2, -2, 1, 0, 0, 1, -2]), n=0..20); # _Wesley Ivan Hurt_, Jul 16 2016

%t PadRight[{}, 100, {2, -2, 1, 0, 0, 1, -2}] (* _Wesley Ivan Hurt_, Jul 16 2016 *)

%o (Magma) &cat [[2, -2, 1, 0, 0, 1, -2]^^20]; // _Wesley Ivan Hurt_, Jul 16 2016

%o (PARI) a(n)=[2, -2, 1, 0, 0, 1, -2][n%7+1] \\ _Charles R Greathouse IV_, Jul 17 2016

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

%K sign,easy

%O 0,1

%A _Wolfdieter Lang_, Feb 27 2014