A327793 The number of nonnegative numbers that can be partitioned into a triangular number (A000217), a square number (A000290), and a pentagonal number (A000326) in n different ways.

1, 2, 10, 12, 13, 10, 23, 25, 16, 36, 31, 34, 27, 45, 36, 50, 68, 61, 53, 68, 57, 72, 60, 59, 61, 87, 85, 88, 82, 97, 91, 106, 95, 98, 127, 93, 111, 125, 127, 124, 109, 127, 152, 122, 114, 146, 147, 132, 157, 169, 118, 180, 156, 158, 163, 168, 180, 178, 190, 184, 187, 196, 207, 191, 210, 204, 207, 206, 190, 227, 231, 203, 195, 219, 264

Offset

1,2

Comments

The number of nonnegative numbers k such that A240088(k) = n.

Links

Table of n, a(n) for n=1..75.

Example

a(0) does not exist since all numbers can be represented as the sum of a triangular, square and pentagonal number;
a(1) = 1 because A240088({0}) = 1;
a(2) = 2 because A240088({3, 18}) = 2;
a(3) = 10 because A240088({1, 2, 4, 8, 9, 13, 14, 35, 98, 168}) = 3;
a(4) = 12 because A240088({5, 6, 7, 21, 25, 30, 34, 39, 43, 48, 63, 78}) = 4;
a(5) = 13 because A240088({10, 11, 12, 17, 20, 23, 28, 33, 69, 193, 203, 230, 243}) = 5;
a(6) = 10 because A240088({19, 24, 32, 44, 53, 55, 74, 90, 111, 130}) = 6;
a(7) = 23 because A240088({15, 16, 27, 29, 40, 46, 56, 60, 62, 68, 73, 84, 85, 95, 108, 113, 123, 135, 139, 163, 165, 273, 553}) = 7; etc.

Mathematica

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];

Crossrefs

Cf. A000217, A000290, A000326, A240088, A275999, A327792.

Sequence in context: A174703 A258081 A300907 * A276103 A101222 A349831

Adjacent sequences: A327790 A327791 A327792 * A327794 A327795 A327796

Keyword

nonn

Author

Robert G. Wilson v, Sep 25 2019

Status

approved