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

0,8

Comments

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

Links

Paolo Xausa, Table of n, a(n) for n = 0..9999

Wikipedia, Triangular number.

Formula

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

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

Example

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.

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}]

Prog

(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

Crossrefs

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

Sequence in context: A368501 A198752 A141462 * A055955 A165071 A341483

Adjacent sequences: A354327 A354328 A354329 * A354331 A354332 A354333

Keyword

nonn,easy

Author

Paolo Xausa, Jun 04 2022

Status

approved