A157926 Coefficients of the first modulo two factor of polynomials of the type: p(x,n)=(x^Prime[n] + 1)/(x + 1).

1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1

Offset

2,1

Comments

Row sums are:

{3, 5, 7, 3, 15, 11, 15, 15, 3, 23, 5, 17, 27, 21, 15,...}.

This procedure gives more than the types with two factors:

Mod[(x^Prime[n] + 1)/(x + 1),2]=f1(x)*f2(x);

In each case the program picks just the first factor.

The classic Golay factorization:

Factor[PolynomialMod[(x^23 + 1)/((x + 1)), 2], Modulus -> 2]

(1 + x + x^5 + x^6 + x^7 + x^9 + x^11)

(1 + x^2 + x^4 + x^5 + x^6 + x^10 + x^11)

The other one mentioned by Sloane and Conway in "Sphere Packings":

Factor[PolynomialMod[(x^47 + 1)/((x + 1)), 2], Modulus -> 2]

(1 + x + x^2 + x^3 + x^5 + x^6 + x^7 + x^9 + x^10 + x^12 + x^13 + x^14 + x^18 + x^19 + x^23)

(1 + x^4 + x^5 + x^9 +x^10 + x^11 + x^13 + x^14 + x^16 + x^17 + x^18 + x^20 +x^21 + x^22 + x^23)

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, pp. 231.

Links

Table of n, a(n) for n=2..106.

Formula

Mod[(x^Prime[n] + 1)/(x + 1),2]=Product[fi(x),{i,0,n}];

Out_(n,m)=coefficients(f1(x)).

Example

{1, 1, 0, 1},
{1, 0, 0, 1, 1, 1, 0, 0, 1},
{1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1},
{1, 0, 1, 0, 0, 1},
{1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1},
{1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1},
{1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1},
{1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1},
{1, 1, 0, 0, 0, 0, 0, 0, 0, 1},
{1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1},
{1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1},
{1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1},
{1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1},
{1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1},
{1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1}

Mathematica

a = Flatten[Table[If[ ( Factor[PolynomialMod[(x^Prime[n] + 1)/((x + 1)), 2], Modulus -> 2] == PolynomialMod[(x^Prime[n] + 1)/((x + 1)), 2]) /. x -> 2, {}, FactorList[PolynomialMod[( x^Prime[n] + 1)/((x + 1)), 2], Modulus -> 2][[2]][[1]]], {n, 2, 30}]];
Table[CoefficientList[a[[n]], x], {n, 1, Length[a]}];
Flatten[%] Table[Apply[Plus, CoefficientList[a[[n]], x]], {n, 1, Length[a]}];

Crossrefs

Sequence in context: A175480 A285568 A014577 * A263243 A131377 A364639

Adjacent sequences: A157923 A157924 A157925 * A157927 A157928 A157929

Keyword

nonn,tabf,uned,obsc

Author

Roger L. Bagula and Gary W. Adamson, Mar 09 2009

Extensions

I started to edit this, changing the definition to read: "Irregular triangle read by rows: factorize (x^prime(n)+1)/(x+1) mod 2, then write down coefficients of lowest degree factor, with exponents in increasing order." This would make a perfectly reasonable sequence, which would begin {1,1}, {1,1,1}, {1,1,1,1,1}, {1,1,0,1}, ... Unfortunately this does not match the terms of the present sequence. Possibly the exponents should be 2^(prime(n)-1) rather than prime(n). I did not investigate this further. - N. J. A. Sloane, Dec 16 2010

Status

approved