A074013 - OEIS

%I #19 Apr 26 2024 14:32:28

%S 0,1,6,20,125,650,3025,15750,78000,390625,1952500,9765625,48831250,

%T 244125000

%N Number of elements of GF(5^n) with trace 1 and subtrace 4.

%C Same as the number of elements of GF(5^n) with trace 2 and subtrace 1. Same as the number of elements of GF(5^n) with trace 3 and subtrace 1. Same as the number of elements of GF(5^n) with trace 4 and subtrace 4.

%H Frank Ruskey, <a href="http://combos.org/TSGF5">Number of Elements of GF(5^n) with given trace and subtrace</a>

%e a(3;3,1)=6. Let GF(5^3) be defined by the field extension GF(5)[x]/( 3+2b+3b^2+b^3 ). The six elements of GF(5^3) with trace 3 and subtrace 1 are { 2+b+b^2, 3+2b+b^2, 4+3b+2b^2, 3+2b+3b^2, 4+3b+4b^2, 4b+4b^2 }.

%o (Sage)

%o def a(n):

%o     ans = 0

%o     for x in GF(5^n):

%o         if x.charpoly().coefficients(sparse=False)[-3:-1]==[4, 1]: ans += 1

%o     return ans  # _Robin Visser_, Apr 26 2024

%Y Cf. A074006, A074007, A074008, A074009, A074010, A074011, A074012.

%K nonn,more

%O 1,3

%A _Frank Ruskey_ and Nate Kube, Aug 19 2002

%E a(8)-a(14) from _Robin Visser_, Apr 26 2024