A214857 Number of triangular numbers in interval [0, n^2].

1, 2, 3, 4, 6, 7, 9, 10, 11, 13, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 33, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 50, 51, 52, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 68, 69, 71, 72, 74, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 93, 95, 96, 98, 99, 100

Offset: 0

Author

Philippe Deléham, Mar 09 2013

Keywords

• nonn
• easy

Comments

Partial sums of A214856.

Examples

			0, 1, 3, 6 are in interval [0, 9], a(3) = 4.
0, 1, 3, 6, 10, 15 are in interval [0, 16], a(4) = 6.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A022846, A214848, A214856.

Programs

Mathematica

nn = 100; tri = Table[n (n + 1)/2, {n, 0, nn}]; Table[Count[tri, ?(# <= n^2 &)], {n, 0, Sqrt[tri[[-1]]]}] (* _T. D. Noe, Mar 11 2013 *)
Table[Floor[(Sqrt[8*n^2+1]-1)/2]+1,{n,0,80}] (* Harvey P. Dale, Oct 14 2014 *)

Python

from math import isqrt
def A214857(n): return isqrt(n**2+1<<3)+1>>1 # Chai Wah Wu, Dec 09 2024

Formula

a(n) = floor((1 + sqrt(1+8*n^2))/2). - Ralf Stephan, Jan 30 2014