The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A309041 Irregular table read by rows: Let P(n,x) be the (monic) minimal polynomial of 2i*cos(Pi/n), where i = sqrt(-1) is the imaginary unit, then a(n,k) = [x^(2k)] P(n,x), n >= 3. 0
 1, 1, 2, 1, 1, 3, 1, 3, 1, 1, 6, 5, 1, 2, 4, 1, 1, 9, 6, 1, 5, 5, 1, 1, 15, 35, 28, 9, 1, 1, 4, 1, 1, 21, 70, 84, 45, 11, 1, 7, 14, 7, 1, 1, 24, 26, 9, 1, 2, 16, 20, 8, 1, 1, 36, 210, 462, 495, 286, 91, 15, 1, 3, 9, 6, 1, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,3 COMMENTS For n >= 3, it is easy to see that [x^(2k+1)] P(n,x) = 0, so they are omitted. Row n (n >= 3) has length A023022(n) + 1 = phi(n)/2 + 1. Let {U(n,x)} be defined as: U(0,x) = 0, U(1,x) = 1, U(n,x) = x*U(n-1,x) + U(n-2,x) for n >= 2, then U(n,x) = Product_{k|n, k>=2} P(k,x) for n > 0, because U(n,x) = Product_{m=1..n-1} (x - 2i*cos(Pi*m/n)) for n > 0. LINKS FORMULA P(n,x) = Product_{0<=m<=n, gcd(m, n)=1} (x - 2i*cos(Pi*m/n)). Equivalently, P(n,x) = Product_{0<=m<=n/2, gcd(m, n)=1} (x^2 + 4*cos(Pi*m/n)) for n != 2. This shows that all terms are positive. P(n,x) = Product_{k|n} U(n/k,x)^mu(k), mu = A008683. Let MPR2(n,x) be the (monic) minimal polynomial of 2*cos(2*Pi/n) as defined in A232624, then: for even n > 2, P(n,x) = MPR2(2n,i*x)*(-1)^A023022(n); for odd n, P(n,x) = MPR2(n,i*x)*MPR2(2n,i*x)*(-1)^A023022(n), i = sqrt(-1). For n > 2, P(n,x) = MPR2(n,-x^2-2)*(-1)^A023022(n). For n > 1, P(n,1) = A061446(n), P(n,2) = A008555(n), P(n,3) = A253807(n), ... For even n > 2, a(n,k) = (-1)^(A023022(n)-k)*A232624(2n,2k). EXAMPLE P(1,x) = x^2 + 4; P(2,x) = x; P(3,x) = x^2 + 1; P(4,x) = x^2 + 2; P(5,x) = x^4 + 3x^2 + 1; P(6,x) = x^2 + 3; P(7,x) = x^6 + 5x^4 + 6x^2 + 1; P(8,x) = x^4 + 4x^2 + 2; P(9,x) = x^6 + 6x^4 + 9x^2 + 1; P(10,x) = x^4 + 5x^2 + 5; ... MATHEMATICA ro[n_] := (P = CoefficientList[p = MinimalPolynomial[2*I*Cos[Pi/n], x], x^2]; P); Flatten[Table[ro[n], {n, 3, 30}]] PROG (PARI) U(n) = sum(i=0, (n-1)/2, binomial(n, 2*i+1)*(poly/2)^(n-2*i-1)*((poly^2+4)/4)^i) P(n) = if(n==1, poly^2+4, my(v=divisors(n)); prod(i=1, #v, U(n/v[i])^moebius(v[i]))) a(n, k) = polcoeff(P(n), 2*k) CROSSREFS Cf. A232624. Cf. P(n,k): A061446 (k=1), A008555 (k=2), A253807 (k=3); Cf. also A023022, A008683. Sequence in context: A050117 A241187 A212822 * A249143 A121560 A326409 Adjacent sequences:  A309038 A309039 A309040 * A309042 A309043 A309044 KEYWORD nonn,tabf AUTHOR Jianing Song, Jul 08 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 7 20:19 EST 2021. Contains 349588 sequences. (Running on oeis4.)