login
A038521
Number of elements of GF(2^n) with trace 1 and subtrace 1.
9
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
0,3
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) = ((-1-i)^(n-2) + (-1+i)^(n-2) + 2^(n-1)) / 2 = 2*A176739(n-2) + A176739(n-3).
a(n) = 2*a(n-2) + 4*a(n-3) for n>3.
(End)
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
MATHEMATICA
LinearRecurrence[{0, 2, 4}, {0, 0, 2, 1}, 33] (* Jean-François Alcover, May 08 2023 *)
PROG
(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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
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
STATUS
approved