A023969 -id:A023969 - cp's OEIS frontend

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

N. J. A. Sloane, Mar 20 2003

nonn

Cf. A000194, A023969, A053188, A056847.

[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

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

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

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)

A080343 a(n) = round(sqrt(2n)) - floor(sqrt(2n)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Mar 20 2003

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A023969, A080352.

from gmpy2 import isqrt_rem
def A080343(n):
    i, j = isqrt_rem(2*n)
    return int(4*(j-i) >= 1) # Chai Wah Wu, Aug 16 2016

Runs are 0^1, 0^1, 0^2 1, 0^2 1, 0^3 1^2, 0^3 1^2, 0^4 1^3, 0^4 1^3, ...
a(n) = 1 iff n >= 4 and n is in the interval [t_k + 1, ..., t_k + floor(k/2)] for some k >= 2, where t_k = k*(k+1)/2 is a triangular number.
a(n) = A023969(2*n). - Michel Marcus, Aug 19 2016

cp's OEIS Frontend

A080344 Partial sums of A023969.

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Formula

A080343 a(n) = round(sqrt(2n)) - floor(sqrt(2n)).

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cp's OEIS Frontend

A080344 Partial sums of A023969.

Views

Author

Keywords

Crossrefs

Programs

Magma

PARI

Python

Formula

A080343 a(n) = round(sqrt(2*n)) - floor(sqrt(2*n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Formula

A080343 a(n) = round(sqrt(2n)) - floor(sqrt(2n)).