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Number of elements of GF(7^n) with trace 1 and subtrace 6.
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%I #11 May 13 2024 05:14:16

%S 0,2,6,56,301,2450,16807,117992,823200,5762400,40356008,282475249

%N Number of elements of GF(7^n) with trace 1 and subtrace 6.

%C Same as the number of elements of GF(7^n) with trace 2 and subtrace 3. Same as the number of elements of GF(7^n) with trace 3 and subtrace 5. Same as the number of elements of GF(7^n) with trace 4 and subtrace 5. Same as the number of elements of GF(7^n) with trace 5 and subtrace 3. Same as the number of elements of GF(7^n) with trace 6 and subtrace 6.

%H Frank Ruskey, <a href="http://combos.org/TSGF7">Number of Elements of GF(7^n) with given trace and subtrace</a>

%e a(3;1,6)=6. Let GF(7^3) be defined by the field extension GF(7)[x]/( 3+b^2+b^3 ). The six elements of GF(7^3) with trace 1 and subtrace 6 are { 1+2b, 2+5b, 1+3b+b^2, 4+5b+b^2, 6+2b+6b^2, 2+4b+6b^2 }.

%o (Sage)

%o def a(n):

%o ans = 0

%o for x in GF(7^n):

%o if x.charpoly().coefficients(sparse=False)[-3:-1]==[6, 1]: ans += 1

%o return ans # _Robin Visser_, May 13 2024

%Y Cf. A074014, A074015, A074016, A074017, A074018, A074019, A074020, A074021, A074022.

%K easy,nonn,more

%O 1,2

%A _Frank Ruskey_ and Nate Kube, Aug 19 2002

%E a(8)-a(12) from _Robin Visser_, May 13 2024