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 A157926 Coefficients of the first modulo two factor of polynomials of the type: p(x,n)=(x^Prime[n] + 1)/(x + 1). 0

%I

%S 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,

%T 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,

%U 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

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

%C Row sums are:

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

%C This procedure gives more than the types with two factors:

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

%C In each case the program picks just the first factor.

%C The classic Golay factorization:

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

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

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

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

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

%C (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)

%C (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)

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

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

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

%e {1, 1, 0, 1},

%e {1, 0, 0, 1, 1, 1, 0, 0, 1},

%e {1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1},

%e {1, 0, 1, 0, 0, 1},

%e {1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1},

%e {1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1},

%e {1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1},

%e {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},

%e {1, 1, 0, 0, 0, 0, 0, 0, 0, 1},

%e {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},

%e {1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1},

%e {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},

%e {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},

%e {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},

%e {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}

%t 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}]];

%t Table[CoefficientList[a[[n]], x], {n, 1, Length[a]}];

%t Flatten[%] Table[Apply[Plus, CoefficientList[a[[n]], x]], {n, 1, Length[a]}];

%K nonn,tabf,uned,obsc,changed

%O 2,1

%A _Roger L. Bagula_ and _Gary W. Adamson_, Mar 09 2009

%E 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

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