cp's OEIS Frontend

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

%I A266498 #41 Jun 18 2025 12:01:39
%S A266498 2,3,4,7,13,22,38,66,115,198,344,595,1031,1786,3093,5357,9279,16071,
%T A266498 27836,48213,83508,144640,250524,433920,751571,1301759,2254713,
%U A266498 3905278,6764140,11715834,20292419,35147501,60877257,105442502,182631770,316327505,547895310,948982514,1643685930,2846947542
%N A266498 Index of the smallest triangular number greater than 3^n.
%C A266498 Also, a(n) is the largest integer m such that binomial(m,2) <= 3^n.
%C A266498 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.
%C A266498 A000217(a(n)) - 3^n = 1 for n = 2 and n = 3. - _Altug Alkan_, Dec 30 2015
%H A266498 I. Bosnjak and R.Tosic, <a href="http://www.dmi.uns.ac.rs/nsjom/Papers/22_1/NSJOM_22_1_133_140.pdf">Some new results concerning two counterfeit coins</a>, Novi Sad Journal of Mathematics 22:1 (1992), 133-140.
%H A266498 T. Khovanova, <a href="http://blog.tanyakhovanova.com/2015/12/two-fake-coins/">Two Fake Coins</a>, 2015.
%H A266498 K. A. Knop and O. B. Polubasov, <a href="http://knop.website/math/ff.pdf">Two counterfeit coins revisited</a>, 2015. (in Russian)
%H A266498 A. Li, <a href="http://dx.doi.org/10.1016/0012-365X(94)90038-8">On the conjecture at two counterfeit coins</a>, Discrete Mathematics 133:1-3 (1994), 301-306.
%F A266498 a(n) = A002024(3^n+1) = A123578(3^n+1).
%F A266498 a(n) = round( sqrt(2*3^n+1/4) ).
%o A266498 (PARI) a(n) = round( sqrt(2*3^n+1/4) );
%o A266498 (Python)
%o A266498 from math import isqrt
%o A266498 def A266498(n): return isqrt(3**n+1<<3)+1>>1 # _Chai Wah Wu_, Jun 18 2025
%Y A266498 Cf. A000217, A000244, A002024, A123578.
%K A266498 nonn
%O A266498 0,1
%A A266498 _Max Alekseyev_, Dec 30 2015