A354329 Triangular number nearest to the sum of the first n positive triangular numbers.

0, 1, 3, 10, 21, 36, 55, 78, 120, 171, 210, 276, 351, 465, 561, 666, 820, 990, 1128, 1326, 1540, 1770, 2016, 2278, 2628, 2926, 3240, 3655, 4095, 4465, 4950, 5460, 5995, 6555, 7140, 7750, 8385, 9180, 9870, 10731, 11476, 12403, 13203, 14196, 15225, 16290, 17205

Offset: 0

Author

Paolo Xausa, Jun 04 2022

Keywords

• nonn
• easy

Examples

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

Links

• Wikipedia, Triangular number.

Crossrefs

Cf. A000217, A000292, A053616, A229118, A354330.

Programs

Mathematica

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

PARI

a(n)=my(t=sqrtint(n*(n+1)*(n+2)/3));(t^2+t)/2;
vector(100,n,a(n-1))

Python

from math import isqrt
def A354329(n): return (m:=isqrt(n*(n*(n + 3) + 2)//3))*(m+1)>>1 # Chai Wah Wu, Jul 15 2022

Formula

a(n) = (t^2+t)/2, where t = floor(sqrt(n*(n+1)*(n+2)/3)).