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A038521 Number of elements of GF(2^n) with trace 1 and subtrace 1.

%I A038521 #48 Nov 12 2024 18:59:21
%S A038521 0,0,2,1,4,10,12,36,64,120,272,496,1024,2080,4032,8256,16384,32640,
%T A038521 65792,130816,262144,524800,1047552,2098176,4194304,8386560,16781312,
%U A038521 33550336,67108864,134225920,268419072,536887296,1073741824,2147450880,4295032832,8589869056
%N A038521 Number of elements of GF(2^n) with trace 1 and subtrace 1.
%H A038521 Colin Barker, <a href="/A038521/b038521.txt">Table of n, a(n) for n = 0..1001</a>
%H A038521 K. Cattel, C. R. Miers, F. Ruskey, J. Sawada, M. Serra, <a href="http://www.ams.org/mathscinet-getitem?mr=2019413">The number of irreducible polynomials over Gf(2) with given trace and subtrace</a>, J. Combin. Math. Combin. Comput. 47 (2003) 31-64.
%H A038521 Frank Ruskey, <a href="http://combos.org/TSpoly">Number of irreducible polynomials over GF(2) with given trace and subtrace</a>
%H A038521 Frank Ruskey, <a href="http://combos.org/TSGF2">Number of elements of GF(2^n) with given trace and subtrace</a>
%H A038521 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,4).
%F A038521 a(n) = C(n, r+0) + C(n, r+4) + C(n, r+8) + ... where r = 3 if n odd, r = 1 if n even.
%F A038521 a(n) = (2^(n-1) - A108520(n-1))/2 if n > 0. - _R. J. Mathar_, Jan 29 2008
%F A038521 From _Colin Barker_, Aug 02 2019: (Start)
%F A038521 G.f.: x^2*(2 + x) / ((1 - 2*x)*(1 + 2*x + 2*x^2)).
%F A038521 a(n) = ((-1-i)^(n-2) + (-1+i)^(n-2) + 2^(n-1)) / 2 = 2*A176739(n-2) + A176739(n-3).
%F A038521 a(n) = 2*a(n-2) + 4*a(n-3) for n>3.
%F A038521 (End)
%p A038521 A038521 := proc(n) local r,a,i ; if n mod 2 = 1 then r := 3 ; else r := 1 ; fi; a :=0 ; for i from r to n by 4 do a := a+binomial(n,i) ; od; a ; end: for n from 0 to 40 do printf("%d,",A038521(n)) ; od: # _R. J. Mathar_, Oct 20 2008
%t A038521 LinearRecurrence[{0, 2, 4}, {0, 0, 2, 1}, 33] (* _Jean-François Alcover_, May 08 2023 *)
%o A038521 (PARI) concat([0, 0], Vec(x*(2 + x) / ((1 - 2*x)*(1 + 2*x + 2*x^2)) + O(x^35))) \\ _Colin Barker_, Aug 02 2019
%o A038521 (Magma) I:=[0,0,2,1]; [m le 4 select I[m] else  2*Self(m-2) + 4*Self(m-3): m in [1..33]]; // _Marius A. Burtea_, Aug 02 2019
%Y A038521 Cf. A038504, A000749, A108520.
%Y A038521 Cf. A038518, A038519, A038520.
%K A038521 easy,nonn
%O A038521 0,3
%A A038521 _Frank Ruskey_
%E A038521 Values duplicated A038520 and were replaced by _R. J. Mathar_, Oct 20 2008
%E A038521 Missing a(0) = 0 inserted by _Andrey Zabolotskiy_, Nov 12 2024