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a(n) = number of polynomials of degree 4 in a regular Groebner basis (graded reverse lexicographic order) of n quadratic polynomials in n variables.
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%I #22 Oct 05 2023 14:33:57

%S 0,0,1,3,5,10,14,22,29,39,50,60,76,91,105,126,146,165,189,215,240,264,

%T 297,329,360,390,430,469,507,544,588,635,681,726,770,826,881,935

%N a(n) = number of polynomials of degree 4 in a regular Groebner basis (graded reverse lexicographic order) of n quadratic polynomials in n variables.

%e For n=3, the leading monomial is x3^4, so a(3) = 1.

%e For n=4, the 3 leading monomials are x1x4^3, x2x4^3, x3x4^3, so a(4) = 3.

%o (Magma)

%o function a(n);

%o F:=GF(251);

%o P<[x]>:=PolynomialRing(F,n,"grevlex");

%o M2:=[ &*[P| x[i] : i in s] : s in Multisets({1..n},2) ];

%o A:=[ &+[Random(F)*m : m in M2] : i in [1..n]];

%o G:=GroebnerBasis(A,4);

%o return #[ g : g in G | TotalDegree(g) eq 4 ];

%o end function;

%Y Cf. A000027 (degree 2), A006463 (degree 3).

%K nonn,more

%O 1,4

%A _Gilles Macario-Rat_, Sep 18 2023