A236043 Number of triangular numbers <= 10^n.

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

1,1

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

Variant of A068092. - R. J. Mathar, Jan 20 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

J. Tanton, Cool Math Newsletter (November 2012)

Formula

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

Example

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

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 *)

Prog

(Python)
def Tri(x):
..count = 0
..for n in range(10**40):
....if n*(n+1)/2 <= 10**x:
......count += 1
....else:
......return count
x = 1
while x < 50:
..print(Tri(x))
..x += 1
(PARI) a236043(n)=floor(sqrt(2*10^n+1/4)+1/2) \\ Chris Boyd, Jan 19 2014
(Magma) [Floor(Sqrt(2*10^n+1/4) + 1/2): n in [1..30]]; // Vincenzo Librandi, Feb 08 2014

Crossrefs

Cf. A000217.

Sequence in context: A197212 A100059 A270062 * A270661 A222908 A270911

Adjacent sequences: A236040 A236041 A236042 * A236044 A236045 A236046

Keyword

nonn,easy

Author

Derek Orr, Jan 18 2014

Extensions

More terms from Jon E. Schoenfield, Feb 07 2014

Status

approved