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%I #15 Oct 30 2019 13:24:24
%S 0,18,168,78,243,130,553,455,515,658,865,945,633,1918,2258,1385,1583,
%T 2828,2135,2335,2785,4533,3168,3478,2790,3868,4193,7328,4953,5278,
%U 6390,8148,8015,4585,9160,10485,7613,12333,12025,10178,9923,9720,12558,11340,17420,11753,14893,16155,16415,14343,18053,19803,16608,27283
%N a(n) is the greatest nonnegative number which has a partition into a triangular number (A000217), a square number (A000290), and a pentagonal number (A000326) in n different ways.
%C The largest nonnegative number k such that A240088(k) = n.
%H Robert G. Wilson v, <a href="/A327792/b327792.txt">Table of n, a(n) for n = 1..200</a>
%e a(0) does not exist since all numbers can be represented as the sum of a triangular, square & pentagonal number.
%t f[n_] := Block[{j, k = 1, lenq, lenr, v = {}, t = PolygonalNumber[3, Range[0, 1 + Sqrt[2 n]]], s = PolygonalNumber[4, Range[0, 1 + Sqrt[n]]], p = PolygonalNumber[5, Range[0, 2 + Sqrt[2 n/3]]]}, u = Select[Union[Join[t, s, p]], # < n + 1 &]; q = IntegerPartitions[n, {3}, u]; lenq = 1 + Length@q; While[k < lenq, j = 1; r = q[[k]]; rr = Permutations@r; lenr = 1 + Length@rr; While[j < lenr, If[ MemberQ[t, rr[[j, 1]]] && MemberQ[s, rr[[j, 2]]] && MemberQ[p, rr[[j, 3]]], AppendTo[v, rr[[j]]]]; j++]; k++]; Length@v];
%Y Cf. A000217, A000290, A000326, A240088, A275999, A327793.
%K nonn
%O 1,2
%A _Robert G. Wilson v_, Sep 25 2019