A236043 Number of triangular numbers <= 10^n.

2, 5, 14, 45, 141, 447, 1414, 4472, 14142, 44721, 141421, 447214, 1414214, 4472136, 14142136, 44721360, 141421356, 447213595, 1414213562, 4472135955, 14142135624, 44721359550, 141421356237, 447213595500, 1414213562373, 4472135955000, 14142135623731

Offset: 0

Author

Derek Orr, Jan 18 2014

Keywords

• nonn
• easy

Comments

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.

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

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

Examples

			There are 4472 triangular numbers less than or equal to 10^7 so a(7) = 4472.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• J. Tanton, Cool Math Newsletter (November 2012)

Crossrefs

Cf. A000217, A002024, A003056. Essentially the same as A068092.

Programs

Magma

[Floor(Sqrt(2*10^n+1/4) + 1/2): n in [1..30]]; // Vincenzo Librandi, Feb 08 2014

Maple

seq(floor(sqrt(2*10^n+1/4)+1/2),n=1..30); # Robert Israel, Dec 22 2024

Mathematica

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 *)

PARI

a236043(n)=floor(sqrt(2*10^n+1/4)+1/2) \\ Chris Boyd, Jan 19 2014

Python

from math import isqrt
def A236043(n): return isqrt(10**n+1<<3)+1>>1 # Chai Wah Wu, Jun 14 2025

Formula

a(n) = floor( sqrt(2*10^n + 1/4) + 1/2 ), adapted from Tanton (see Links section). - Chris Boyd, Jan 19 2014

a(n) = A068092(n + 1) for n >= 2. - R. J. Mathar, Jan 20 2014

a(n) = A003056(10^n) + 1 = A002024(10^n + 1). - Andrew Howroyd, Dec 21 2024

Extensions

More terms from Jon E. Schoenfield, Feb 07 2014

a(0) prepended by Andrew Howroyd, Dec 21 2024