Search: id:A157926
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%I A157926
%S A157926 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,
%T A157926 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,
%U A157926 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 A157926 Coefficients of the first modulo two factor of polynomials of the trtype: 
               p(x,n)=(x^Prime[n] + 1)/(x + 1).
%C A157926 Row sums are:
%C A157926 {3, 5, 7, 3, 15, 11, 15, 15, 3, 23, 5, 17, 27, 21, 15,...}.
%C A157926 This procedure gives more than the types with two factors:
%C A157926 Mod[(x^Prime[n] + 1)/(x + 1),2]=f1(x)*f2(x);
%C A157926 In each case the program picks just the first factor.
%C A157926 The classic Golay factorization:
%C A157926 Factor[PolynomialMod[(x^23 + 1)/((x + 1)), 2], Modulus -> 2]
%C A157926 (1 + x + x^5 + x^6 + x^7 + x^9 + x^11)
%C A157926 (1 + x^2 + x^4 + x^5 + x^6 + x^10 + x^11)
%C A157926 The other one mentioned by Sloane and Conway in "Sphere Packings":
%C A157926 Factor[PolynomialMod[(x^47 + 1)/((x + 1)), 2], Modulus -> 2]
%C A157926 (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 A157926 (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 A157926 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, pp. 231.
%F A157926 Mod[(x^Prime[n] + 1)/(x + 1),2]=Product[fi(x),{i,0,n}];
%F A157926 Out_(n,m)=coefficients(f1(x)).
%e A157926 {1, 1, 0, 1},
%e A157926 {1, 0, 0, 1, 1, 1, 0, 0, 1},
%e A157926 {1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1},
%e A157926 {1, 0, 1, 0, 0, 1},
%e A157926 {1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1},
%e A157926 {1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1},
%e A157926 {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 A157926 {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 A157926 {1, 1, 0, 0, 0, 0, 0, 0, 0, 1},
%e A157926 {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 A157926 {1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1},
%e A157926 {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 A157926 {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 A157926 {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 A157926 {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 A157926 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 A157926 Table[CoefficientList[a[[n]], x], {n, 1, Length[a]}];
%t A157926 Flatten[%] Table[Apply[Plus, CoefficientList[a[[n]], x]], {n, 1, Length[a]}];
%Y A157926 Sequence in context: A100672 A079559 A014577 this_sequence A131377 A077049 
               A124895
%Y A157926 Adjacent sequences: A157923 A157924 A157925 this_sequence A157927 A157928 
               A157929
%K A157926 nonn,tabl
%O A157926 2,1
%A A157926 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Mar 
               09 2009

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