A074004 Number of elements of GF(3^n) with trace 1 and subtrace 1.

0, 1, 3, 12, 21, 81, 252, 729, 2187, 6480, 19845, 59049, 176904, 531441, 1594323, 4785156, 14344533, 43046721, 129146724, 387420489, 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 1.

Examples

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

Links

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

Crossrefs

Cf. A074000, A074001, A074002, A074003, A074005.

Programs

Sage

def a(n):
    ans = 0
    for x in GF(3^n):
        if x.charpoly().coefficients(sparse=False)[-3:-1]==[1, 1]: ans += 1
    return ans  # Robin Visser, Dec 28 2024

Extensions

a(14) corrected, unverified terms a(17)-a(20) removed. Based on the original Data in A074000-A074005, a(17)-a(20) are possibly equal to 14344533, 43046721, 129146724, 387420489. - Andrey Zabolotskiy, Nov 11 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