A074005 Number of elements of GF(3^n) with trace 1 and subtrace 2.

0, 2, 3, 6, 30, 81, 252, 702, 2187, 6642, 19602, 59049, 176904, 532170, 1594323, 4780782, 14351094, 43046721, 129146724, 387400806, 1162261467

Offset: 1

Author

Frank Ruskey and Nate Kube, Aug 19 2002

Keywords

• nonn
• more

Comments

Same as the number of elements of GF(3^n) with trace 2 and subtrace 2.

Examples

			a(2;1,2)=2. Let GF(3^2) be defined by the field extension GF(3)[x]/( 2+b+b^2 ). The two elements of GF(3^2) with trace 1 and subtrace 2 are { 1+b, 2b }.
		

Links

• Frank Ruskey, Number of Elements of GF(3^n) with given trace and subtrace

Crossrefs

Cf. A074000, A074001, A074002, A074003, A074004.

Programs

SageMath

d = {(0, 0): [1], (0, 1): [0], (0, 2): [0], (1, 0): [1], (1, 1): [0], (1, 2): [0], (2, 0): [1], (2, 1): [0], (2, 2): [0]}
for n in (2..9):
    for a in d.values(): a.append(0)
    k. = GF((3, n))
    for x in k:
        d[(x.trace(), x.charpoly().list()[-3])][-1] += 1
print(d[(1, 2)]) # Andrey Zabolotskiy, Nov 07 2024

Extensions

a(9) and a(14)-a(15) corrected, unverified terms a(17)-a(20) removed. Based on the original Data in A074000-A074005, a(17)-a(20) are possibly equal to 14351094, 43046721, 129146724, 387400806. - Andrey Zabolotskiy, Nov 07 2024

Terms a(17)-a(20) recomputed and added again (verified that all the terms a(17)-a(20) conjectured by Andrey Zabolotskiy are correct), and added term a(21). - Robin Visser, Dec 28 2024