A006463 -id:A006463 - cp's OEIS frontend

A347737 Zero together with the partial sums of A238005.

0, 0, 0, 0, 1, 1, 2, 3, 5, 5, 7, 9, 11, 13, 15, 16, 20, 23, 25, 28, 31, 33, 37, 41, 45, 48, 52, 54, 59, 64, 67, 72, 78, 81, 86, 89, 94, 100, 106, 110, 116, 122, 126, 132, 138, 141, 148, 155, 162, 168, 174, 179, 186, 193, 198, 204, 212, 218, 226, 234, 240, 248, 256, 260

Offset: 0

Omar E. Pol, Sep 11 2021

nonn

a(n) is also the total number of ones in the first n rows of A347579, n >= 1.

a(n) is also the total number of zeros in the first n rows of the triangles A196020, A211343, A231345, A236106, A237048 (simpler), A239662, A261699, A271344, A272026, A280850, A285574, A285891, A285914, A286013, A296508 (and possibly others), n >= 1.

Cf. A001227, A003056, A006463, A060831, A237593, A238005, A347579.

Accumulate@Table[Length@Select[Select[IntegerPartitions@n,DuplicateFreeQ],Differences@MinMax@#=={Length@#}&],{n,60}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

from math import isqrt
def A347737(n): return (r:=isqrt((n+1<<3)+1)-1>>1)*(6*n+4-r*(r+3))//6-((t:=isqrt(m:=n>>1))+(s:=isqrt(n)))*(t-s)-(sum(n//k for k in range(1,s+1))-sum(m//k for k in range(1,t+1))<<1) # Chai Wah Wu, Jun 07 2025

a(n) = A006463(n+1) - A060831(n).

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.

Original entry on oeis.org

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

Gilles Macario-Rat, Sep 18 2023

			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.

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

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;

cp's OEIS Frontend

A347737 Zero together with the partial sums of A238005.

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