This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A214857 #21 Dec 09 2024 15:54:26
%S A214857 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,
%T A214857 35,37,38,40,41,42,44,45,47,48,50,51,52,54,55,57,58,59,61,62,64,65,66,
%U A214857 68,69,71,72,74,75,76,78,79,81,82,83,85,86,88,89,91,92,93,95,96,98,99,100
%N A214857 Number of triangular numbers in interval [0, n^2].
%C A214857 Partial sums of A214856.
%H A214857 Alois P. Heinz, <a href="/A214857/b214857.txt">Table of n, a(n) for n = 0..10000</a>
%F A214857 a(n) = floor((1 + sqrt(1+8*n^2))/2). - _Ralf Stephan_, Jan 30 2014
%e A214857 0, 1, 3, 6 are in interval [0, 9], a(3) = 4.
%e A214857 0, 1, 3, 6, 10, 15 are in interval [0, 16], a(4) = 6.
%t A214857 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 *)
%t A214857 Table[Floor[(Sqrt[8*n^2+1]-1)/2]+1,{n,0,80}] (* _Harvey P. Dale_, Oct 14 2014 *)
%o A214857 (Python)
%o A214857 from math import isqrt
%o A214857 def A214857(n): return isqrt(n**2+1<<3)+1>>1 # _Chai Wah Wu_, Dec 09 2024
%Y A214857 Cf. A022846, A214848, A214856.
%K A214857 nonn,easy
%O A214857 0,2
%A A214857 _Philippe Deléham_, Mar 09 2013