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

Author

Greyson C. Wesley, Jul 04 2022

Keywords

• nonn
• easy

Examples

			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.
		

Links

• 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).

Programs

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 *)

Formula

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

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