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A236043
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Number of triangular numbers <= 10^n.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..200
J. Tanton, Cool Math Newsletter (November 2012)
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FORMULA
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a(n) = floor( sqrt(2*10^n + 1/4) + 1/2 ), adapted from Tanton (see Links section). - Chris Boyd, Jan 19 2014
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EXAMPLE
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There are 4472 triangular numbers less than or equal to 10^7 so a(7) = 4472.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000217.
Sequence in context: A197212 A100059 A270062 * A270661 A222908 A270911
Adjacent sequences: A236040 A236041 A236042 * A236044 A236045 A236046
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KEYWORD
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nonn,easy
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AUTHOR
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Derek Orr, Jan 18 2014
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EXTENSIONS
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More terms from Jon E. Schoenfield, Feb 07 2014
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STATUS
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approved
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