A355510 a(n) is the number of monic polynomials of degree n over GF(7) without linear factors.

0, 0, 21, 112, 819, 5712, 39991, 279936, 1959552, 13716864, 96018048, 672126336, 4704884352, 32934190464, 230539333248, 1613775332736, 11296427329152, 79074991304064, 553524939128448, 3874674573899136, 27122722017293952

Offset

0,3

Links

Table of n, a(n) for n=0..20.

A. Knopfmacher and J. Knopfmacher, Counting irreducible factors of polynomials over a finite field, Discrete Mathematics, Volume 112, Issues 1-3, 1993, Pages 103-118.

Index entries for linear recurrences with constant coefficients, signature (7).

Formula

O.g.f.: (1 - z)^7/(1 - 7*z) - 1.

For n >= 7, a(n) = 6^7 * 7^(n-7).

Example

a(0) = 0 since all constant polynomials are units (as GF(7) is a field).
a(1) = 0 since all polynomials of degree 1 have linear factors.
a(2), the number of quadratic polynomials without linear factors, then coincides with the number of irreducible quadratics in GF(7), which is known to be M(7,2), where M(a,d) is the necklace polynomial, so a(2) = 21.

Mathematica

CoefficientList[Series[(1-z)^7/(1-7 z)-1, {z, 0, 15}]//Normal, z] (* For all terms *)
(7^(#-7)) &/@ Range[7, 15]*6^7 (* For n>=7 *)
Join[{0, 0, 21, 112, 819, 5712, 39991}, NestList[7#&, 279936, 20]] (* Harvey P. Dale, Oct 29 2022 *)

Crossrefs

Sequence in context: A255285 A157265 A275916 * A129135 A158091 A121628

Adjacent sequences: A355507 A355508 A355509 * A355511 A355512 A355513

Keyword

nonn,easy

Author

Greyson C. Wesley, Jul 04 2022

Status

approved