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 A236043 #49 Jun 14 2025 18:34:21
%S A236043 2,5,14,45,141,447,1414,4472,14142,44721,141421,447214,1414214,
%T A236043 4472136,14142136,44721360,141421356,447213595,1414213562,4472135955,
%U A236043 14142135624,44721359550,141421356237,447213595500,1414213562373,4472135955000,14142135623731
%N A236043 Number of triangular numbers <= 10^n.
%C A236043 Except for 5, all numbers begin with either a 4 or a 1. If strictly less than, the 5 would become a 4, satisfying this conjecture.
%C A236043 This is not a conjecture, it is a fact and it is the result from the square root of 2 and 20 times powers of ten. - _Robert G. Wilson v_, Jan 11 2015
%C A236043 Tanton (2012) discusses the equivalent sequence based on excluding zero from the triangular numbers, and presents the relevant formula, which, being asymptotic to floor[sqrt(2*10^n)], explains the observation in the first comment. - _Chris Boyd_, Jan 19 2014
%H A236043 Vincenzo Librandi, <a href="/A236043/b236043.txt">Table of n, a(n) for n = 0..200</a>
%H A236043 J. Tanton, <a href="http://www.jamestanton.com/wp-content/uploads/2012/03/Cool-Math-Newsletter_November2012.pdf">Cool Math Newsletter (November 2012)</a>
%F A236043 a(n) = floor( sqrt(2*10^n + 1/4) + 1/2 ), adapted from Tanton (see Links section). - _Chris Boyd_, Jan 19 2014
%F A236043 a(n) = A068092(n + 1) for n >= 2. - _R. J. Mathar_, Jan 20 2014
%F A236043 a(n) = A003056(10^n) + 1 = A002024(10^n + 1). - _Andrew Howroyd_, Dec 21 2024
%e A236043 There are 4472 triangular numbers less than or equal to 10^7 so a(7) = 4472.
%p A236043 seq(floor(sqrt(2*10^n+1/4)+1/2),n=1..30); # _Robert Israel_, Dec 22 2024
%t A236043 Table[ Floor[ Sqrt[2*10^n + 1] + 1/2], {n, 25}] (* _Vincenzo Librandi_, Feb 08 2014; modified by _Robert G. Wilson v_, Jan 11 2015 *)
%o A236043 (PARI) a236043(n)=floor(sqrt(2*10^n+1/4)+1/2) \\ _Chris Boyd_, Jan 19 2014
%o A236043 (Magma) [Floor(Sqrt(2*10^n+1/4) + 1/2): n in [1..30]]; // _Vincenzo Librandi_, Feb 08 2014
%o A236043 (Python)
%o A236043 from math import isqrt
%o A236043 def A236043(n): return isqrt(10**n+1<<3)+1>>1 # _Chai Wah Wu_, Jun 14 2025
%Y A236043 Cf. A000217, A002024, A003056. Essentially the same as A068092.
%K A236043 nonn,easy
%O A236043 0,1
%A A236043 _Derek Orr_, Jan 18 2014
%E A236043 More terms from _Jon E. Schoenfield_, Feb 07 2014
%E A236043 a(0) prepended by _Andrew Howroyd_, Dec 21 2024