A266498 Index of the smallest triangular number greater than 3^n.

2, 3, 4, 7, 13, 22, 38, 66, 115, 198, 344, 595, 1031, 1786, 3093, 5357, 9279, 16071, 27836, 48213, 83508, 144640, 250524, 433920, 751571, 1301759, 2254713, 3905278, 6764140, 11715834, 20292419, 35147501, 60877257, 105442502, 182631770, 316327505, 547895310, 948982514, 1643685930, 2846947542

Offset: 0

Author

Max Alekseyev, Dec 30 2015

Keywords

• nonn

Comments

Also, a(n) is the largest integer m such that binomial(m,2) <= 3^n.

a(n) gives a theoretical upper bound for the number of coins such that two fake coins (of equal weight lighter than the other coins) among them can be identified in n weightings on a balance scale. It was shown that the bound is achievable for all n<=10, but it remains an open question if the bound is achievable for n>10.

A000217(a(n)) - 3^n = 1 for n = 2 and n = 3. - Altug Alkan, Dec 30 2015

Links

• I. Bosnjak and R.Tosic, Some new results concerning two counterfeit coins, Novi Sad Journal of Mathematics 22:1 (1992), 133-140.
• T. Khovanova, Two Fake Coins, 2015.
• K. A. Knop and O. B. Polubasov, Two counterfeit coins revisited, 2015. (in Russian)
• A. Li, On the conjecture at two counterfeit coins, Discrete Mathematics 133:1-3 (1994), 301-306.

Crossrefs

Cf. A000217, A000244, A002024, A123578.

Programs

PARI

a(n) = round( sqrt(2*3^n+1/4) );

Python

from math import isqrt
def A266498(n): return isqrt(3**n+1<<3)+1>>1 # Chai Wah Wu, Jun 18 2025

Formula

a(n) = A002024(3^n+1) = A123578(3^n+1).

a(n) = round( sqrt(2*3^n+1/4) ).