A080344 Partial sums of A023969.

0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 4, 5, 6, 6, 6, 6, 6, 6, 7, 8, 9, 10, 10, 10, 10, 10, 10, 10, 11, 12, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 18, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 23, 24, 25, 26, 27, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 29, 30, 31, 32, 33, 34, 35

Offset: 0

Author

N. J. A. Sloane, Mar 20 2003

Keywords

• nonn

Crossrefs

Cf. A000194, A023969, A053188, A056847.

Programs

Magma

[1/2*(n+1-Floor(Sqrt(n+1)+1/2)-Abs(n+1-(Floor(Sqrt(n+1)+1/2))^2)):n in [0..90]]; // Marius A. Burtea, May 09 2019

PARI

f(n) = sqrtint(4*n)-2*sqrtint(n); \\ A023969
a(n) = sum(k=0, n, f(k)); \\ Michel Marcus, May 10 2019

Python

from math import isqrt
def A080344(n): return n+1-(k:=(m:=isqrt(n+1))+int(n>=m*(m+1)))-abs(n+1-k**2)>>1 # Chai Wah Wu, Jun 05 2025

Formula

From Ridouane Oudra, May 11 2019: (Start)

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

a(n) = (1/2)*(n + 1 - A000194(n+1) - abs(n + 1 - A000194(n+1)^2)).

a(n) = (1/2)*(A056847(n+1) - A053188(n+1)). (End)