In 1830, Legendre (1979) proved that every number larger than 1791 is a sum of four hexagonal numbers, and Duke and Schulze-Pillot (1990) improved this to three hexagonal numbers for every sufficiently large integer.
There are exactly 13 positive integers that cannot be represented using four hexagonal numbers, namely 5, 10, 11, 20, 25, 26, 38, 39, 54, 65, 70, 114, and 130 (OEIS A007527; Guy 1994a).
Similarly, there are only two positive integers that cannot be represented using five hexagonal numbers, namely:
Every positive integer can be represented using six hexagonal numbers.
Duke, W. and Schulze-Pillot, R. "Representations of Integers by Positive Ternary Quadratic Forms and Equidistribution of Lattice Points
on Ellipsoids." Invent. Math.99, 49-57, 1990.Guy,
R. K. "Every Number Is Expressible as the Sum of How Many Polygonal Numbers?."
Amer. Math. Monthly101, 169-172, 1994a.Guy, R. K.
"Sums of Squares." §C20 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136-138,
1994b.Legendre, A.-M. Théorie des nombres, 4th ed., 2 vols.
Paris: A. Blanchard, 1979.Sloane, N. J. A. Sequences
A000384/M4108 and A007527/M3739
in "The On-Line Encyclopedia of Integer Sequences."