A123578 -id:A123578 - cp's OEIS frontend

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

Max Alekseyev, Dec 30 2015

nonn

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

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.

Cf. A000217, A000244, A002024, A123578.

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

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

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

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

A126862 Numbers k that have a component C(1,1) when expanded in the binomial basis of order t=3.

Original entry on oeis.org

3, 6, 8, 12, 14, 17, 22, 24, 27, 31, 37, 39, 42, 46, 51, 58, 60, 63, 67, 72, 78, 86, 88, 91, 95, 100, 106, 113, 122, 124, 127, 131, 136, 142, 149, 157, 167, 169, 172, 176, 181, 187, 194, 202, 211, 222, 224, 227, 231, 236, 242, 249, 257, 266, 276, 288, 290, 293, 297

Offset: 1

R. J. Mathar, Mar 15 2007

Each positive integer k has a unique binomial expansion k = C(k_t,t) + C(k_{t-1},t-1) + ... + C(k_v,v) for a given order t, where k_t > k_{t-1} > ... > k_v >= v >= 1. The sequence contains those k for which v=1 and k_v=1 at t=3. The equivalent sequence for t=2 is A000124.

			Expansions in t=3 for k=19 up to 23 are k=19=C(5,3)+C(4,2)+C(3,1);
k=20=C(6,3); k=21=C(6,3)+C(2,2); k=22=C(6,3)+C(2,2)+C(1,1); k=23=C(6,3)+C(3,2).
Of these, only k=22 has a C(1,1) component and makes it into the sequence.

Cf. A123578.

With[{res = Map[ResourceFunction["BinomialNumberSystemTriplet"], Range[300]]},Position[res[[All, 1]], 1] // Flatten] (* Shenghui Yang, Jul 31 2025 *)

cp's OEIS Frontend

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

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