A210576 Positive integers that cannot be expressed as sum of one or more nontrivial binomial coefficients.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29

Offset: 1

Author

Douglas Latimer, Mar 22 2012

Keywords

• nonn
• fini
• full

Comments

The nontrivial binomial coefficients are C(n,k), 2 <= k <= n-2 (A006987).

I conjectured that the sequence is finite, consisting of the terms listed.

This conjecture is now proved. - Douglas Latimer, Apr 10 2013

Note that this sequence allows the same binomial coefficient to be used multiple times. - T. D. Noe, Apr 12 2013

These are the only values of the angular momentum for which a wavefunction with such an angular momentum and the symmetry of a dodecahedron is impossible. [Baez] - Andrey Zabolotskiy, Mar 28 2018

Examples

			The smallest terms in the sequence are 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14 because 6, 10 and 15 cannot be terms, as these are the lowest nontrivial binomial coefficients; 12 and 16 cannot be terms, as these are the lowest sums of two nontrivial binomial coefficients; and sums of three or more nontrivial binomial coefficients cannot exclude any of the listed terms.
		

Links

• John Baez, Quantum Mechanics and the Dodecahedron, Dec 31 2017.
• Douglas Latimer, Computation of Terms <= 30.
• Douglas Latimer, Terms Listed Are the Entire Sequence.

Crossrefs

A210578 contains many of the integers that cannot be elements of this sequence.

Cf. A006987 and A007318.

Positions of zeros in A008651. Cf. A005796.