A329598 Partial sums of the nontriangular numbers (A014132).

2, 6, 11, 18, 26, 35, 46, 58, 71, 85, 101, 118, 136, 155, 175, 197, 220, 244, 269, 295, 322, 351, 381, 412, 444, 477, 511, 546, 583, 621, 660, 700, 741, 783, 826, 870, 916, 963, 1011, 1060, 1110, 1161, 1213, 1266, 1320, 1376, 1433, 1491, 1550, 1610, 1671, 1733

Offset

1,1

Comments

Terms which are triangular: 6, 136, 351, 741, 2415, 3916, 5995, 12561, 17391, 23436, ..., .

Links

Table of n, a(n) for n=1..52.

Formula

a(n) = Sum_{i=1..n} A014132(i).

a(n) = A000217(n) + A060432(n). [corrected by Gerald Hillier, Jul 31 2022]

Example

The nontriangular numbers begin 2, 4, 5, 7, ..., so their partial sums begin 2, 6, 11, 18, etc.

Mathematica

triQ[n_] := IntegerQ @ Sqrt[8n + 1]; Accumulate@ Select[ Range@ 70, !triQ@# &]

Prog

(Python)
from math import isqrt
def A329598(n): return (k:=(r:=isqrt(m:=n+1<<1))+int((m<<2)>(r<<2)*(r+1)+1)-1)*(k*(-k - 3) + 6*n - 2)//6 + (n*(n+3)>>1) # Chai Wah Wu, Jun 18 2024

Crossrefs

Cf. A329472, A014132, A086849.

Cf. A000217, A060432.

Sequence in context: A039745 A298872 A298875 * A248469 A331952 A225386

Adjacent sequences: A329595 A329596 A329597 * A329599 A329600 A329601

Keyword

nonn

Author

Metin Sariyar and Robert G. Wilson v, Nov 17 2019

Status

approved