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

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

Author

R. J. Mathar, Mar 15 2007

Keywords

• easy
• nonn

Comments

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.

Examples

			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.
		

Crossrefs

Cf. A123578.

Programs

Mathematica

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