A068092 Index of smallest triangular number with n digits.

1, 4, 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

Author

Amarnath Murthy, Feb 19 2002

Keywords

• base
• easy
• nonn

Comments

Look at the interleaving of the decimal expansion of the square roots of 2 and 20.

Examples

			a(4) = 45 as the 45th triangular number is 45*46/2 = 1035 while the 44th is 990.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A002024, A002193, A010476.

Cf. A068093, A068094, A095863, A095864, A095865, A095866.

Programs

Magma

[Round(Sqrt(2*10^(n-1))) : n in [1..30]]; // Vincenzo Librandi, Oct 05 2011

Mathematica

f[n_] := Block[{a = Floor[Sqrt[2*10^n]]}, If[a(a + 1)/2 < 10^n, a++ ]; Return[a]]; Table[ f[n], {n, 0, 30} ]

PARI

a(n) = round(sqrt(2*10^(n-1))) \\ Charles R Greathouse IV, Oct 04 2011

Python

from math import isqrt
def A068092(n): return isqrt(10**(n-1)<<3)+1>>1 # Chai Wah Wu, Oct 17 2022

Formula

a(n) = b where b = floor(sqrt(2*10^(n-1))) and if b(b+1)/2 < 10^(n-1), then b = b+1. [corrected by Ray Chandler, Oct 04 2011]

a(n) = round((2*10^(n-1))^(1/2)). - Vladeta Jovovic, Mar 08 2004

a(n) = A002024(10^(n-1)). - Michel Marcus, Jan 27 2022

Extensions

Edited and extended by Robert G. Wilson v, Feb 21 2002