A038521 Number of elements of GF(2^n) with trace 1 and subtrace 1.
0, 0, 2, 1, 4, 10, 12, 36, 64, 120, 272, 496, 1024, 2080, 4032, 8256, 16384, 32640, 65792, 130816, 262144, 524800, 1047552, 2098176, 4194304, 8386560, 16781312, 33550336, 67108864, 134225920, 268419072, 536887296, 1073741824, 2147450880, 4295032832, 8589869056
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1001
- K. Cattel, C. R. Miers, F. Ruskey, J. Sawada, M. Serra, The number of irreducible polynomials over Gf(2) with given trace and subtrace, J. Combin. Math. Combin. Comput. 47 (2003) 31-64.
- Frank Ruskey, Number of irreducible polynomials over GF(2) with given trace and subtrace
- Frank Ruskey, Number of elements of GF(2^n) with given trace and subtrace
- Index entries for linear recurrences with constant coefficients, signature (0,2,4).
Programs
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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
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Maple
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
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Mathematica
LinearRecurrence[{0, 2, 4}, {0, 0, 2, 1}, 33] (* Jean-François Alcover, May 08 2023 *) -
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
Formula
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.
a(n) = (2^(n-1) - A108520(n-1))/2 if n > 0. - R. J. Mathar, Jan 29 2008
From Colin Barker, Aug 02 2019: (Start)
G.f.: x^2*(2 + x) / ((1 - 2*x)*(1 + 2*x + 2*x^2)).
a(n) = 2*a(n-2) + 4*a(n-3) for n>3.
(End)
Extensions
Values duplicated A038520 and were replaced by R. J. Mathar, Oct 20 2008
Missing a(0) = 0 inserted by Andrey Zabolotskiy, Nov 12 2024