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

0,3

Links

Table of n, a(n) for n=0..46.

Wikipedia, Triangular number.

Formula

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A194141 A281153 A014105 * A146012 A027917 A038347

Adjacent sequences: A354326 A354327 A354328 * A354330 A354331 A354332

Keyword

nonn,easy

Author

Paolo Xausa, Jun 04 2022

Status

approved