A214856 -id:A214856 - cp's OEIS frontend

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

Philippe Deléham, Mar 09 2013

Partial sums of A214856.

			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.

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

Cf. A022846, A214848, A214856.

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 *)

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

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

A378301 a(n) is the number of triangular numbers (A000217) in the interval [n^2, (n + 1)^2].

Original entry on oeis.org

2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1

Offset: 0

Ctibor O. Zizka, Nov 22 2024

nonn

After n=22, is the frequency of 1 always > 2? Are terms only 1 or 2? - Bill McEachen, Dec 10 2024

The distinct terms of this sequence are only 1 and 2. The asymptotic densities of the occurrences of 1 and 2 are 2-sqrt(2) = 0.585... and sqrt(2)-1 = 0.414..., respectively. The asymptotic mean of this sequence is sqrt(2). - Amiram Eldar, Dec 11 2024

			n = 0: in the interval [0, 1] are 2 triangular numbers {0, 1}, thus a(0) = 2.
n = 1: in the interval [1, 4] are 2 triangular numbers {1, 3}, thus a(1) = 2.

Cf. A000217, A000290, A001110, A002024, A003056, A214856, A214857.

s[n_] := Floor[(Sqrt[8*n+1]-1)/2]; a[n_] := s[(n + 1)^2] - s[n^2 - 1]; a[0] = 2; Array[a, 100, 0] (* Amiram Eldar, Dec 09 2024 *)

a(n) = sum(k=n^2, (n+1)^2, ispolygonal(k,3)); \\ Michel Marcus, Dec 09 2024

from math import isqrt
def A378301(n): return -(isqrt(m:=n**2<<3)+1>>1)+(isqrt(m+(n+1<<4))+1>>1) # Chai Wah Wu, Dec 09 2024

a(n) = 2 for n from A001110.
a(n) = A003056((n+1)^2) - A003056(n^2-1) for n >= 1. - Amiram Eldar, Dec 09 2024
a(n) = A002024((n+1)^2+1) - A002024(n^2). - Chai Wah Wu, Dec 09 2024

a(53) corrected by Michel Marcus, Dec 09 2024

cp's OEIS Frontend

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

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