A309945 a(n) = floor(n - sqrt(2*n-1)).

0, 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 60

Offset: 1

Author

Zhandos Mambetaliyev, Aug 24 2019

Keywords

• nonn

Comments

The subsequence consisting of numbers that appear twice is A007590.

Sequence as triangle:

0;

0;

0; 1, 2;

2, 3, 4;

4, 5, 6, 7, 8;

8, 9, 10, 11, 12;

12, 13, 14, 15, 16, 17, 18;

18, 19, 20, 21, 22, 23, 24;

...

a(1) = 0; for n > 1, a(n) is the number of squares strictly between 2*n - 2 and n^2.

Examples

			For n = 3, 2*n - 2 = 4, n^2 = 9, no square numbers strictly between 4 and 9, a(3) = 0.
For n=5, 2*n - 2 = 8, n^2 = 25, two square numbers (9, 16) strictly between 8 and 25, a(5) = 2.
		

Links

• Table of n, a(n) for n = 1..999

Crossrefs

Cf. A007590, A080476, A016813.

Programs

Mathematica

Table[Floor[n-(2*n-1)^(1/2)],{n,73}] (* Stefano Spezia, Aug 24 2019 *)

PARI

a(n) = floor(n - sqrt(2*n-1)); \\ Jinyuan Wang, Aug 26 2019

Python

from math import isqrt
def A309945(n): return (m:=n-1)-isqrt(m<<1) # Chai Wah Wu, Aug 04 2022

Formula

a(n) = n-1-floor(sqrt(2*n-2)). - Wesley Ivan Hurt, Dec 03 2020