cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A042979 Number of degree-n irreducible polynomials over GF(2) with trace = 0 and subtrace = 1.

Original entry on oeis.org

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
Offset: 1

Views

Author

Frank Ruskey

Keywords

Links

Crossrefs

Cf. A042980, A042981, A042982.
Cf. A074027, A074028, A074029, A074030.

Programs

  • 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 *)
  • 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))
/* Joerg Arndt, Jun 28 2012 */

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).

A042980 Number of degree-n irreducible polynomials over GF(2) with trace = 0 and subtrace = 0.

Original entry on oeis.org

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
Offset: 1

Views

Author

Frank Ruskey

Keywords

Links

Crossrefs

Cf. A042979, A042981, A042982.
Cf. A074027, A074028, A074029, A074030.

Programs

  • 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}];
Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Jun 28 2012, from formula *)
  • 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))
/* Joerg Arndt, Jun 28 2012 */

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)}.

A042982 Number of degree-n irreducible polynomials over GF(2) with trace = 1 and subtrace = 1.

Original entry on oeis.org

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
Offset: 1

Views

Author

Frank Ruskey

Keywords

Links

Crossrefs

Cf. A042979, A042980, A042981.
Cf. A074027, A074028, A074029, A074030.

Programs

  • 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 *)
  • 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 */

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)}.

A074027 Number of binary Lyndon words of length n with trace 0 and subtrace 0 over Z_2.

Original entry on oeis.org

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
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 21 2002

Keywords

Comments

Same as the number of binary Lyndon words of length n with trace 0 and subtrace 0 over GF(2).

Examples

			a(6;0,0)=2 since the two binary Lyndon words of trace 0, subtrace 0 and length 6 are { 001111, 010111 }.

Links

Crossrefs

Cf. A074028, A074029, A074030.

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.

Extensions

Terms a(33) onward from Max Alekseyev, Apr 09 2013

A074029 Number of binary Lyndon words of length n with trace 1 and subtrace 0 over Z_2.

Original entry on oeis.org

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
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 21 2002

Keywords

Comments

Same as the number of binary Lyndon words of length n with trace 1 and subtrace 0 over GF(2).

Examples

			a(3;1,0)=1 since the one binary Lyndon word of trace 1, subtrace 0 and length 3 is { 001 }.

Links

Crossrefs

Cf. A074027, A074028, A074030.

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.

Extensions

Terms a(33) onward from Max Alekseyev, Apr 09 2013

A074030 Number of binary Lyndon words of length n with trace 1 and subtrace 1 over Z_2.

Original entry on oeis.org

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
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 21 2002

Keywords

Comments

Same as the number of binary Lyndon words of length n with trace 1 and subtrace 1 over GF(2).

Links

Crossrefs

Cf. A074027, A074028, A074029.

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.

Extensions

Corrected by Franklin T. Adams-Watters, Oct 25 2006
Terms a(33) onward from Max Alekseyev, Apr 09 2013

A042981 Number of degree-n irreducible polynomials over GF(2) with trace = 1 and subtrace = 0.

Original entry on oeis.org

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
Offset: 1

Views

Author

Frank Ruskey

Keywords

Links

Crossrefs

Cf. A042979, A042980, A042982.
Cf. A074027, A074028, A074029, A074030.

Programs

  • 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}]
(* Jean-François Alcover, Jun 28 2012, from formula *)
  • 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))
/* Joerg Arndt, Jun 28 2012 */

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)}.
Showing 1-7 of 7 results.

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