|
| |
|
|
A042982
|
|
Number of degree-n irreducible polynomials over GF(2) with trace = 1 and subtrace = 1.
|
|
7
|
|
|
|
0, 1, 0, 1, 2, 2, 5, 8, 13, 27, 45, 85, 160, 288, 550, 1024, 1920, 3654, 6885, 13107, 24989, 47616, 91225, 174760, 335462, 645435, 1242600, 2396745, 4628480, 8947294, 17318945, 33554432, 65074253, 126324495, 245424829, 477218560, 928645120, 1808400384, 3524082400, 6871947672, 13408665600, 26178873147
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,5
|
|
|
LINKS
|
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
K. Cattell, C. R. Miers, F. Ruskey, J. Sawada and M. Serra, The Number of Irreducible Polynomials over GF(2) with Given Trace and Subtrace, J. Comb. Math. and Comb. Comp., 47 (2003) 31-64.
F. Ruskey, Number of irreducible polynomials over GF(2) with given trace and subtrace
|
|
|
FORMULA
|
a(n) = (1/n) * Sum_{ L(n, k) : n+k = 3 mod 4}, where L(n, k) = Sum_{ mu(d)*binomial(n/d, k/d) : d|gcd(n, k)}.
|
|
|
MATHEMATICA
|
L[n_, k_] := Sum[ MoebiusMu[d]*Binomial[n/d, k/d], {d, Divisors[GCD[n, k]]}]/n; a[n_] := Sum[ If[ Mod[n+k, 4] == 3, L[n, k], 0], {k, 0, n}]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Jun 28 2012, from formula *)
|
|
|
PROG
|
(PARI)
L(n, k) = sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%4==3, L(n, k), 0 ) ) / n;
vector(33, n, a(n))
/* Joerg Arndt, Jun 28 2012 */
|
|
|
CROSSREFS
|
Cf. A042979, A042980, A042981.
Cf. A074027, A074028, A074029, A074030.
Sequence in context: A293674 A052527 A335443 * A340249 A006367 A246807
Adjacent sequences: A042979 A042980 A042981 * A042983 A042984 A042985
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Frank Ruskey
|
|
|
STATUS
|
approved
|
| |
|
|
