A269483 a(n) = n^12 - n^11 + n^9 - n^8 + n^6 - n^4 + n^3 - n + 1.

1, 1, 2359, 368089, 12783421, 196890121, 1822428931, 11898664849, 60247241209, 251393376241, 900900990991, 2855262053161, 8177824843189, 21515718297529, 52663539957211, 121132473843361, 263947231891441, 548461977100129

Offset: 0

Author

Ilya Gutkovskiy, Feb 27 2016

Keywords

• nonn
• easy

Comments

a(n) = Phi_21(n) where Phi_k(x) is the k-th cyclotomic polynomial.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)
• Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
• Index to values of cyclotomic polynomials of integer argument
• Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Crossrefs

Cf. similar sequences of the type Phi_k(n) listed in A269442.

Programs

GAP

List([0..20], n-> n^12-n^11+n^9-n^8+n^6-n^4+n^3-n+1); # G. C. Greubel, Apr 24 2019

Magma

[n^12-n^11+n^9-n^8+n^6-n^4+n^3-n+1: n in [0..20]]; // Vincenzo Librandi, Feb 28 2016

Mathematica

Table[Cyclotomic[21, n], {n, 0, 17}]
CoefficientList[Series[(1 -12x +2424x^2 +337214x^3 +8182695x^4 [phone] x^5 [phone]x^6 [phone]x^7 [phone]x^8 [phone]x^9 + 612600 x^10 +5406x^11 +x^12)/(1-x)^13, {x, 0, 33}], x] (* Vincenzo Librandi, Feb 28 2016 *)

PARI

a(n) = polcyclo(21, n); \\ Michel Marcus, Feb 29 2016

Python

A269483_list, m = [], [479001600, -2674425600, 6386688000, -8501915520, 6889478400, -3482100720, 1080164160, -194177280, 17948256, -666714, 5418, 0, 1]
for _ in range(10**2):
    A269483_list.append(m[-1])
    for i in range(12):
        m[i+1] += m[i] # Chai Wah Wu, Feb 28 2016

Sage

[n^12-n^11+n^9-n^8+n^6-n^4+n^3-n+1 for n in (0..20)] # G. C. Greubel, Apr 24 2019

Formula

G.f.: (1 - 12*x + 2424*x^2 + 337214*x^3 + 8182695*x^4 + 58741344*x^5 + 156377856*x^6 + 168607380*x^7 + 73943271*x^8 + 12191420*x^9 + 612600*x^10 + 5406*x^11 + x^12)/(1-x)^13.

Sum_{n>=0} 1/a(n) = 2.00042670913...