Number of times that the sum of 4 unordered triangular numbers equals n.

1, 1, 1, 2, 2, 1, 3, 2, 2, 3, 3, 2, 4, 4, 2, 4, 4, 3, 5, 4, 3, 5, 6, 4, 6, 4, 4, 6, 6, 4, 6, 8, 5, 7, 6, 4, 8, 8, 6, 6, 8, 6, 8, 8, 6, 9, 9, 6, 10, 9, 6, 10, 10, 6, 8, 10, 7, 11, 13, 8, 9, 10, 10, 10, 10, 7, 13, 14, 9, 10, 10, 10, 13, 14, 8, 10, 15, 10, 14, 12

0

This is the case a = b = c = d = 1 in the paper by Wang and Sun.

T. D. Noe, Plot of 1001 terms

T. D. Noe, Table of 1001 terms

Min Wang and Zhi-Hong Sun, On the number of representations of n by a*x(x−1)/2 + b*y(y−1)/2 + c*z(z−1)/2 + d*t(t−1)/2, arXiv 1507.03485 (Jul 13 2015)

(Mma) nn = 12; tri[n_] := n*(n+1)/2; t = Flatten[Table[tri[i] + tri[j] + tri[k] + tri[l], {i, 0, nn}, {j, 0, i}, {k, 0, j}, {l, 0, k}]]; t2 = Take[Tally[Sort[t]], tri[nn] + 1]; Transpose[t2][[2]]

Cf. A002636.

nonn,nice

T. D. Noe, Jul 20 2015