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%I #6 May 16 2014 15:50:04
%S 1,1,2,1,2,3,3,4,1,4,3,4,2,2,5,3,5,3,5,4,5,7,3,4,4,6,6,4,6,3,5,7,6,4,
%T 1,7,7,6,5,6,9,5,7,7,8,6,8,4,6,6,7,9,4,10,3,6,9,7,8,5,9,7,6,7,5,11,9,
%U 7,3,7,12,13,7,7,6,9,11,6,11,8,7,10,10,8,8,8,11,5,8,5,8,11,10,10,6,14,10,6,7,7
%N Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an even square (A016742) and a generalized pentagonal number (A001318).
%C It is conjectured (1.1) and then proved by theorem 1.2 that all positive integers can be so represented [Sun, pp. 4-5].
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/0905.0635">On Universal Sums Of Polygonal Numbers</a>, arXiv:0905.0635v18 [math.NT] 26 Oct 2011
%t planeFigurative[n_, r_] := pf[n, r] = (n - 2) Binomial[r, 2] + r; s = Sort@ Table[ planeFigurative[3, i] + planeFigurative[3, j] + planeFigurative[3, k], {i, 0, 14}, {j, 0, 10, 2}, {k, -8, 8}]; Table[ Count[s, n], {n, 0, 50}]
%Y Cf. A002636, A115288, A160324, A160325, A160326, A240088, A242442.
%K nonn,easy
%O 1,3
%A _Robert G. Wilson v_, May 14 2014