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

Author

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

Keywords

• nonn

Comments

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

Examples

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

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.

Crossrefs

Cf. A329472, A014132, A086849.

Cf. A000217, A060432.

Programs

Mathematica

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

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

Formula

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

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