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Number of degree-n irreducible polynomials over GF(2) with trace = 0 and subtrace = 1.
+10
11
0, 0, 1, 0, 2, 2, 4, 8, 13, 24, 48, 80, 160, 288, 541, 1024, 1920, 3626, 6912, 13056, 24989, 47616, 91136, 174760, 335462, 645120, 1242904, 2396160, 4628480, 8947294, 17317888, 33554432, 65074253, 126320640, 245428574, 477211280, 928645120, 1808400384, 3524068955, 6871947672, 13408665600, 26178823218
FORMULA
a(n) = (1/n) * Sum_{k=0..n, n+k == 0 (mod 4)} L(n, k), where L(n, k) = Sum_{d|gcd(n, k)} mu(d)*binomial(n/d, k/d).
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] == 0, 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==0, L(n, k), 0 ) ) / n;
vector(33, n, a(n))
Number of degree-n irreducible polynomials over GF(2) with trace = 0 and subtrace = 0.
+10
8
1, 0, 0, 1, 1, 2, 5, 6, 15, 24, 45, 85, 155, 288, 550, 1008, 1935, 3626, 6885, 13107, 24940, 47616, 91225, 174590, 335626, 645120, 1242600, 2396745, 4627915, 8947294, 17318945, 33552384, 65076240, 126320640, 245424829, 477218560, 928638035, 1808400384, 3524082400, 6871921458, 13408691175, 26178823218
FORMULA
a(n) = (1/n) * Sum_{ L(n, k) : n+k = 2 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]==2, L[n, k], 0], {k, 0, n}];
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==2, L(n, k), 0 ) ) / n;
vector(33, n, a(n))
Number of degree-n irreducible polynomials over GF(2) with trace = 1 and subtrace = 1.
+10
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
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))
Number of binary Lyndon words of length n with trace 0 and subtrace 0 over Z_2.
+10
7
1, 0, 0, 0, 1, 2, 5, 8, 15, 24, 45, 80, 155, 288, 550, 1024, 1935, 3626, 6885, 13056, 24940, 47616, 91225, 174760, 335626, 645120, 1242600, 2396160, 4627915, 8947294, 17318945, 33554432, 65076240, 126320640, 245424829, 477211280, 928638035, 1808400384, 3524082400
COMMENTS
Same as the number of binary Lyndon words of length n with trace 0 and subtrace 0 over GF(2).
FORMULA
a(2n) = A042979(2n), a(2n+1) = A042980(2n+1). This follows from Cattell et al. (see A042979), Main Theorem on p. 33 and Theorem 4 on p. 44.
EXAMPLE
a(6;0,0)=2 since the two binary Lyndon words of trace 0, subtrace 0 and length 6 are { 001111, 010111 }.
Number of binary Lyndon words of length n with trace 1 and subtrace 0 over Z_2.
+10
7
1, 1, 1, 1, 1, 2, 4, 8, 15, 27, 48, 85, 155, 288, 541, 1024, 1935, 3654, 6912, 13107, 24940, 47616, 91136, 174760, 335626, 645435, 1242904, 2396745, 4627915, 8947294, 17317888, 33554432, 65076240, 126324495, 245428574, 477218560, 928638035, 1808400384, 3524068955
COMMENTS
Same as the number of binary Lyndon words of length n with trace 1 and subtrace 0 over GF(2).
FORMULA
a(2n) = A042982(2n), a(2n+1) = A049281(2n+1). This follows from Cattell et al. (see A042979), Main Theorem on p. 33 and Theorem 4 on p. 44.
EXAMPLE
a(3;1,0)=1 since the one binary Lyndon word of trace 1, subtrace 0 and length 3 is { 001 }.
Number of binary Lyndon words of length n with trace 1 and subtrace 1 over Z_2.
+10
7
0, 0, 0, 1, 2, 3, 5, 8, 13, 24, 45, 85, 160, 297, 550, 1024, 1920, 3626, 6885, 13107, 24989, 47709, 91225, 174760, 335462, 645120, 1242600, 2396745, 4628480, 8948385, 17318945, 33554432, 65074253, 126320640, 245424829, 477218560, 928645120, 1808414181, 3524082400
COMMENTS
Same as the number of binary Lyndon words of length n with trace 1 and subtrace 1 over GF(2).
FORMULA
a(2n) = A042981(2n), a(2n+1) = A042982(2n+1). This follows from Cattell et al. (see A042979), Main Theorem on p. 33 and Theorem 4 on p. 44.
Number of degree-n irreducible polynomials over GF(2) with trace = 1 and subtrace = 0.
+10
6
1, 0, 1, 1, 1, 3, 4, 8, 15, 24, 48, 85, 155, 297, 541, 1024, 1935, 3626, 6912, 13107, 24940, 47709, 91136, 174760, 335626, 645120, 1242904, 2396745, 4627915, 8948385, 17317888, 33554432, 65076240, 126320640, 245428574, 477218560, 928638035, 1808414181, 3524068955, 6871947672, 13408691175, 26178823218
FORMULA
a(n) = (1/n) * Sum_{ L(n, k) : n+k = 1 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] == 1, L[n, k], 0], {k, 0, n}];
Table[a[n], {n, 1, 32}]
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==1, L(n, k), 0 ) ) / n;
vector(33, n, a(n))
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