A365763 a(n) = number of polynomials of degree 4 in a regular Groebner basis (graded reverse lexicographic order) of n quadratic polynomials in n variables.
0, 0, 1, 3, 5, 10, 14, 22, 29, 39, 50, 60, 76, 91, 105, 126, 146, 165, 189, 215, 240, 264, 297, 329, 360, 390, 430, 469, 507, 544, 588, 635, 681, 726, 770, 826, 881, 935
Offset: 1
Examples
For n=3, the leading monomial is x3^4, so a(3) = 1. For n=4, the 3 leading monomials are x1x4^3, x2x4^3, x3x4^3, so a(4) = 3.
Programs
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Magma
function a(n); F:=GF(251); P<[x]>:=PolynomialRing(F,n,"grevlex"); M2:=[ &*[P| x[i] : i in s] : s in Multisets({1..n},2) ]; A:=[ &+[Random(F)*m : m in M2] : i in [1..n]]; G:=GroebnerBasis(A,4); return #[ g : g in G | TotalDegree(g) eq 4 ]; end function;