The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A165912 Number of alternating polynomials of degree 3n in GF(2)[X], n>0. 2
 2, 0, 2, 2, 4, 6, 12, 20, 38, 66, 124, 224, 420, 774, 1456, 2720, 5140, 9690, 18396, 34918, 66576, 127038, 243148, 465920, 894784, 1720530, 3314018, 6390930, 12341860, 23860200, 46182444, 89477120, 173534032, 336857610, 654471204, 1272578048, 2476377540, 4822410222, 9397535280 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS We define an alternating polynomial as follows: let I be the set of irreducible polynomials of degree > 1 over GF(2) and Sym_3 the symmetric group on 3 elements. For a polynomial P in I of degree n, we define P*(X) = X^n P(1/X) and P+(X) = P(X+1). The operators define an action of the group Sym_3 over I. Then an alternating polynomial is defined by the property that P*=P+. The degree of an alternating polynomial is always 0 mod 3. The numbers in the sequence are always even. These polynomials are invariant under the action of the alternating subgroup Alt_3 of S3. LINKS G. C. Greubel, Table of n, a(n) for n = 1..3300 J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174 FORMULA a(n) = 2*(sum_{d|n, n/d != 0 mod 3} mu(n/d)*(2^d - (-1)^d))/(3n). a(n) = 2 * A165920(n). MATHEMATICA a[n_] := 2*DivisorSum[n, Boole[Mod[n/#, 3] != 0] MoebiusMu[n/#]*(2^# - (-1)^#) &]/(3 n); Array[a, 40] (* Jean-François Alcover, Dec 03 2015, adapted from PARI *) 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)%3!=0, L(n, k), 0 ) ) / n; vector(55, n, a(n)) /* Joerg Arndt, Jun 28 2012 */ CROSSREFS A001037 is the enumeration by degree of the polynomials of the set I. A000048 is the enumeration by degree of the polynomials such that P=P* (self-reciprocal polynomials) which is the same as the one for the polynomials such that P=P+ or P=((P+)*)+. Sequence in context: A331262 A300822 A118658 * A301823 A301999 A171936 Adjacent sequences: A165909 A165910 A165911 * A165913 A165914 A165915 KEYWORD easy,nonn AUTHOR Jean Francis Michon and Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009 EXTENSIONS Edited by N. J. A. Sloane, May 15 2010 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 05:37 EST 2023. Contains 367575 sequences. (Running on oeis4.)