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A355510 a(n) is the number of monic polynomials of degree n over GF(7) without linear factors. 1
0, 0, 21, 112, 819, 5712, 39991, 279936, 1959552, 13716864, 96018048, 672126336, 4704884352, 32934190464, 230539333248, 1613775332736, 11296427329152, 79074991304064, 553524939128448, 3874674573899136, 27122722017293952 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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.
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
KEYWORD
nonn,easy
AUTHOR
Greyson C. Wesley, Jul 04 2022
STATUS
approved

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Last modified May 25 16:30 EDT 2024. Contains 372801 sequences. (Running on oeis4.)