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

0,8

Links

Table of n, a(n) for n=0..79.

Formula

a(n) = (A056847(n+1) - A053188(n+1))/2. - Ridouane Oudra, May 11 2019

Prog

(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

Crossrefs

Cf. A000194, A023969, A053188, A056847.

Sequence in context: A196183 A200264 A104406 * A194225 A025782 A120506

Adjacent sequences: A080341 A080342 A080343 * A080345 A080346 A080347

Keyword

nonn

Author

N. J. A. Sloane, Mar 20 2003

Status

approved