login
Partial sums of the nontriangular numbers (A014132).
1

%I #30 Jun 18 2024 01:52:20

%S 2,6,11,18,26,35,46,58,71,85,101,118,136,155,175,197,220,244,269,295,

%T 322,351,381,412,444,477,511,546,583,621,660,700,741,783,826,870,916,

%U 963,1011,1060,1110,1161,1213,1266,1320,1376,1433,1491,1550,1610,1671,1733

%N Partial sums of the nontriangular numbers (A014132).

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

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

%F a(n) = A000217(n) + A060432(n). [corrected by _Gerald Hillier_, Jul 31 2022]

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

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

%o (Python)

%o from math import isqrt

%o 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

%Y Cf. A329472, A014132, A086849.

%Y Cf. A000217, A060432.

%K nonn

%O 1,1

%A _Metin Sariyar_ and _Robert G. Wilson v_, Nov 17 2019