cp's OEIS Frontend

It is conjectured (1.1) and then proved by theorem 1.2 that all positive integers can be so represented [Sun, pp. 4-5].

a(n) = SUM(A010054(k)*A052343(n-k): 0<=k<=n);

A002636(n) <= a(n).

Partial sums of smallest number representable as the sum of 3 triangular numbers in exactly n ways. The subsequence of triangular numbers in the partial sum begins: 3, 15, 36. The subsequence of primes in the partial sum begins: 3, 1531, 11887, 17737, 37997, 43441.

Fermat asserted, and Gauss proved, that every number is the sum of three triangular numbers (cf. A002636).

For exactly half of the integers k in 1..11898, decomposing k into triangular numbers by the greedy algorithm (i.e., letting T1 be the largest triangular number <= k, then letting T2 be the largest triangular number <= k-T1, etc.) yields a decomposition of k into three or fewer positive triangular numbers, but for any K > 11898, the greedy algorithm decomposes more than half of the integers k in 1..K into four or more positive triangular numbers.

Even an approach that assigns only T1 "greedily" but allows T2 to be any triangular number will usually not yield a set of three triangular numbers whose sum is k: for more than half of the integers k in 1..K for any K > 40304762, no such sum exists in which the largest of the three triangular numbers is the largest triangular number <= k. The smallest such k is a(1)=20 (see Example section).

For some values of k, there exists no set of three triangular numbers summing to k unless the largest of the three is neither T(m(k)) nor T(m(k)-1); the smallest of these is a(2)=50, for which a solution to T(m(k)-2) + T2 + T3 = k does exist (see Example section).