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 A236043 Number of triangular numbers <= 10^n. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified January 27 04:11 EST 2021. Contains 340443 sequences. (Running on oeis4.)