A038518 Number of elements of GF(2^n) with trace 0 and subtrace 0.

0, 1, 1, 1, 6, 6, 16, 36, 56, 136, 256, 496, 1056, 2016, 4096, 8256, 16256, 32896, 65536, 130816, 262656, 523776, 1048576, 2098176, 4192256, 8390656, 16777216, 33550336, 67117056, 134209536, 268435456, 536887296, 1073709056, 2147516416

Offset

0,5

Links

Colin Barker, Table of n, a(n) for n = 0..1000

F. Ruskey, Number of irreducible polynomials over GF(2) with given trace and subtrace

F. 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).

Formula

C(n, r+0)+C(n, r+4)+C(n, r+8)+... where r = 0 if n odd, r = 2 if n even.

G.f.: (-x^3+x^2+x)/[(1-2x)(1+2x+2x^2)].

a(0)=0; a(n) = ( 2^n - (-1-i)^n - (-1+i)^n )/4, i=sqrt(-1). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004

a(n) = 2*a(n-2) + 4*a(n-3) for n>3. - Colin Barker, Aug 02 2019

Maple

0, seq(1/4*2^k-1/4*(-1-I)^k-1/4*(-1+I)^k, k=1..40); seq(coeff(convert(series((-x^3+x^2+x)/((1-2*x)*(1+2*x+2*x^2)), x, 50), polynom), x, i), i=0..40); # C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004

Mathematica

LinearRecurrence[{0, 2, 4}, {0, 1, 1, 1}, 40] (* Harvey P. Dale, Mar 31 2020 *)

Prog

(PARI) concat(0, Vec(x*(1 + x - x^2) / ((1 - 2*x)*(1 + 2*x + 2*x^2)) + O(x^40))) \\ Colin Barker, Aug 02 2019

Crossrefs

Cf. A038503, A038505.

Cf. A038519, A038520, A038521.

Sequence in context: A315813 A168460 A315814 * A151724 A335795 A315815

Adjacent sequences: A038515 A038516 A038517 * A038519 A038520 A038521

Keyword

easy,nonn

Author

Frank Ruskey

Status

approved