A354330 Distance from the sum of the first n positive triangular numbers to the nearest triangular number.

0, 0, 1, 0, 1, 1, 1, 6, 0, 6, 10, 10, 13, 10, 1, 14, 4, 21, 12, 4, 0, 1, 8, 22, 28, 1, 36, 1, 35, 30, 10, 4, 11, 10, 0, 20, 51, 41, 10, 71, 4, 62, 41, 6, 45, 75, 91, 88, 97, 85, 55, 10, 51, 100, 10, 99, 20, 124, 29, 56, 130, 90, 48, 20, 7, 10, 30, 68, 125, 136

Offset: 0

Author

Paolo Xausa, Jun 04 2022

Keywords

• nonn
• easy

Comments

a(n) = 0 for n in {0, 1, 3, 8, 20, 34} = A224421.

Examples

			a(4) = 1 because the sum of the first 4 positive triangular numbers is 1 + 3 + 6 + 10 = 20, the nearest triangular number is 21 and 21 - 20 = 1.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..9999
• Wikipedia, Triangular number.

Crossrefs

Cf. A000217, A000292, A053616, A224421, A238455, A351830, A354329.

Programs

Mathematica

nterms=100;Table[ts=n(n+1)(n+2)/3;t=Floor[Sqrt[ts]];Abs[t^2+t-ts]/2,{n,0,nterms-1}]

PARI

a(n)=my(ts=n*(n+1)*(n+2)/3,t=sqrtint(ts));abs(t^2+t-ts)/2;
vector(100,n,a(n-1)) \\ Paolo Xausa, Jul 06 2022

Python

from math import isqrt
def A354330(n): return abs((m:=isqrt(k:=n*(n*(n + 3) + 2)//3))*(m+1)-k)>>1 # Chai Wah Wu, Jul 15 2022

Formula

a(n) = A053616(A000292(n)).

a(n) = abs(A000292(n) - A354329(n)).