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

Author

Omar E. Pol, Sep 11 2021

Keywords

• nonn

Comments

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

Crossrefs

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.

Programs

Mathematica

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

Python

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

Formula

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