Representing every integer 1-1000 as a sum of three triangular numbers, without using the nonessential triangulars: {8,11,16,17,23,24,29,31,36,38,41,43,45,49,50,59,60,61,62,65}

<<DiscreteMath`Combinatorica`

Take[Union[Map[Flatten[{Apply[Plus,Map[(#+#^2)/2 &,#]],Sort[#]}]&,Strings[Complement[Range[0,66],
{8,11,16,17,23,24,29,31,36,38,41,43,45,49,50,59,60,61,62,65}],3]]],3860]

{{0,0,0,0}, {1,0,0,1}, {2,0,1,1}, {3,0,0,2}, {3,1,1,1}, {4,0,1,2}, {5,1,1,2}, {6,0,0,3}, {6,0,2,2}, {7,0,1,3}, {7,1,2,2}, {8,1,1,3}, {9,0,2,3}, {9,2,2,2}, {10,0,0,4}, {10,1,2,3}, {11,0,1,4}, {12,0,3,3}, {12,1,1,4}, {12,2,2,3}, {13,0,2,4}, {13,1,3,3}, {14,1,2,4}, {15,0,0,5}, {15,2,3,3}, {16,0,1,5}, {16,0,3,4}, {16,2,2,4}, {17,1,1,5}, {17,1,3,4}, {18,0,2,5}, {18,3,3,3}, {19,1,2,5}, {19,2,3,4}, {20,0,4,4}, {21,0,0,6}, {21,0,3,5}, {21,1,4,4}, {21,2,2,5}, {22,0,1,6}, {22,1,3,5}, {22,3,3,4}, {23,1,1,6}, {23,2,4,4}, {24,0,2,6}, {24,2,3,5}, {25,0,4,5}, {25,1,2,6}, {26,1,4,5}, {26,3,4,4}, {27,0,3,6}, {27,2,2,6}, {27,3,3,5}, {28,0,0,7}, {28,1,3,6}, {28,2,4,5}, {29,0,1,7}, {30,0,5,5}, {30,1,1,7}, {30,2,3,6}, {30,4,4,4}, {31,0,2,7}, {31,0,4,6}, {31,1,5,5}, {31,3,4,5}, {32,1,2,7}, {32,1,4,6}, {33,2,5,5}, {33,3,3,6}, {34,0,3,7}, {34,2,2,7}, {34,2,4,6}, {35,1,3,7}, {35,4,4,5}, {36,0,5,6}, {36,3,5,5}, {37,1,5,6}, {37,2,3,7}, {37,3,4,6}, {38,0,4,7}, {39,1,4,7}, {39,2,5,6}, {40,3,3,7}, {40,4,5,5}, {41,2,4,7}, {41,4,4,6}, {42,0,6,6}, {42,3,5,6}, {43,0,5,7}, {43,1,6,6}, {44,1,5,7}, {44,3,4,7}, {45,0,0,9}, {45,2,6,6}, {45,5,5,5}, {46,0,1,9}, {46,2,5,7}, {46,4,5,6}, {47,1,1,9}, {48,0,2,9}, {48,3,6,6}, {48,4,4,7}, {49,0,6,7}, {49,1,2,9}, {49,3,5,7}, {50,1,6,7}, {51,0,3,9}, {51,2,2,9}, {51,5,5,6}, {52,1,3,9}, {52,2,6,7}, {52,4,6,6}, {53,4,5,7}, {54,2,3,9}, {55,0,0,10}, {55,0,4,9}, {55,3,6,7}, {56,0,1,10}, {56,0,7,7}, {56,1,4,9}, {57,1,1,10}, {57,1,7,7}, {57,3,3,9}, {57,5,6,6}, {58,0,2,10}, {58,2,4,9}, {58,5,5,7}, {59,1,2,10}, {59,2,7,7}, {59,4,6,7}, {60,0,5,9}, {61,0,3,10}, {61,1,5,9}, {61,2,2,10}, {61,3,4,9}, {62,1,3,10}, {62,3,7,7}, {63,2,5,9}, {63,6,6,6}, {64,2,3,10}, {64,5,6,7}, {65,0,4,10}, {65,4,4,9}, {66,0,6,9}, {66,1,4,10}, {66,3,5,9}, {66,4,7,7}, {67,1,6,9}, {67,3,3,10}, {68,2,4,10}, {69,2,6,9}, {70,0,5,10}, {70,4,5,9}, {70,6,6,7}, {71,1,5,10}, {71,3,4,10}, {71,5,7,7}, {72,3,6,9}, {73,0,7,9}, {73,2,5,10}, {74,1,7,9}, {75,4,4,10}, {75,5,5,9}, {76,0,6,10}, {76,2,7,9}, {76,3,5,10}, {76,4,6,9}, {77,1,6,10}, {77,6,7,7}, {78,0,0,12}, {79,0,1,12}, {79,2,6,10}, {79,3,7,9}, {80,1,1,12}, {80,4,5,10}, {81,0,2,12}, {81,5,6,9}, {82,1,2,12}, {82,3,6,10}, {83,0,7,10}, {83,4,7,9}, {84,0,3,12}, {84,1,7,10}, {84,2,2,12}, {84,7,7,7}, {85,1,3,12}, {85,5,5,10}, {86,2,7,10}, {86,4,6,10}, {87,2,3,12}, {87,6,6,9}, {88,0,4,12}, {88,5,7,9}, {89,1,4,12}, {89,3,7,10}, {90,0,9,9}, {90,3,3,12}, {91,0,0,13}, {91,1,9,9}, {91,2,4,12}, {91,5,6,10}, {92,0,1,13}, {93,0,5,12}, {93,1,1,13}, {93,2,9,9}, {93,4,7,10}, {94,0,2,13}, {94,1,5,12}, {94,3,4,12}, {94,6,7,9}, {95,1,2,13}, {96,2,5,12}, {96,3,9,9}, {97,0,3,13}, {97,2,2,13}, {97,6,6,10}, {98,1,3,13}, {98,4,4,12}, {98,5,7,10}, {99,0,6,12}, {99,3,5,12}, {100,0,9,10}, {100,1,6,12}, {100,2,3,13}, {100,4,9,9}, {101,0,4,13}, {101,1,9,10}, {101,7,7,9}, {102,1,4,13}, {102,2,6,12}, {103,2,9,10}, {103,3,3,13}, {103,4,5,12}, {104,2,4,13}, {104,6,7,10}, {105,0,0,14}, {105,3,6,12}, {105,5,9,9}, {106,0,1,14}, {106,0,5,13}, {106,0,7,12}, {106,3,9,10}, {107,1,1,14}, {107,1,5,13}, {107,1,7,12}, {107,3,4,13}, {108,0,2,14}, {108,5,5,12}, {109,1,2,14}, {109,2,5,13}, {109,2,7,12}, {109,4,6,12}, {110,0,10,10}, {110,4,9,10}, {111,0,3,14}, {111,1,10,10}, {111,2,2,14}, {111,4,4,13}, {111,6,9,9}, {111,7,7,10}, {112,0,6,13}, {112,1,3,14}, {112,3,5,13}, {112,3,7,12}, {113,1,6,13}, {113,2,10,10}, {114,2,3,14}, {114,5,6,12}, {115,0,4,14}, {115,2,6,13}, {115,5,9,10}, {116,1,4,14}, {116,3,10,10}, {116,4,5,13}, {116,4,7,12}, {117,3,3,14}, {118,2,4,14}, {118,3,6,13}, {118,7,9,9}, {119,0,7,13}, {120,0,0,15}, {120,0,5,14}, {120,1,7,13}, {120,4,10,10}, {120,6,6,12}, {121,0,1,15}, {121,1,5,14}, {121,3,4,14}, {121,5,5,13}, {121,5,7,12}, {121,6,9,10}, {122,1,1,15}, {122,2,7,13}, {122,4,6,13}, {123,0,2,15}, {123,0,9,12}, {123,2,5,14}, {124,1,2,15}, {124,1,9,12}, {125,3,7,13}, {125,4,4,14}, {125,5,10,10}, {126,0,3,15}, {126,0,6,14}, {126,2,2,15}, {126,2,9,12}, {126,3,5,14}, {127,1,3,15}, {127,1,6,14}, {127,5,6,13}, {127,6,7,12}, {128,7,9,10}, {129,2,3,15}, {129,2,6,14}, {129,3,9,12}, {129,4,7,13}, {130,0,4,15}, {130,4,5,14}, {131,1,4,15}, {131,6,10,10}, {132,3,3,15}, {132,3,6,14}, {133,0,7,14}, {133,0,10,12}, {133,2,4,15}, {133,4,9,12}, {133,6,6,13}, {134,1,7,14}, {134,1,10,12}, {134,5,7,13}, {134,7,7,12}, {135,0,5,15}, {135,5,5,14}, {135,9,9,9}, {136,0,9,13}, {136,1,5,15}, {136,2,7,14}, {136,2,10,12}, {136,3,4,15}, {136,4,6,14}, {137,1,9,13}, {138,2,5,15}, {138,5,9,12}, {138,7,10,10}, {139,2,9,13}, {139,3,7,14}, {139,3,10,12}, {140,4,4,15}, {140,6,7,13}, {141,0,6,15}, {141,3,5,15}, {141,5,6,14}, {142,1,6,15}, {142,3,9,13}, {143,4,7,14}, {143,4,10,12}, {144,2,6,15}, {144,6,9,12}, {145,4,5,15}, {145,9,9,10}, {146,0,10,13}, {146,4,9,13}, {147,1,10,13}, {147,3,6,15}, {147,6,6,14}, {147,7,7,13}, {148,0,7,15}, {148,5,7,14}, {148,5,10,12}, {149,1,7,15}, {149,2,10,13}, {150,0,9,14}, {150,5,5,15}, {151,1,9,14}, {151,2,7,15}, {151,4,6,15}, {151,5,9,13}, {151,7,9,12}, {152,3,10,13}, {153,2,9,14}, {154,3,7,15}, {154,6,7,14}, {154,6,10,12}, {155,9,10,10}, {156,0,12,12}, {156,3,9,14}, {156,4,10,13}, {156,5,6,15}, {157,1,12,12}, {157,6,9,13}, {158,4,7,15}, {159,2,12,12}, {160,0,10,14}, {160,4,9,14}, {161,1,10,14}, {161,5,10,13}, {161,7,7,14}, {161,7,10,12}, {162,3,12,12}, {162,6,6,15}, {163,2,10,14}, {163,5,7,15}, {164,7,9,13}, {165,0,9,15}, {165,5,9,14}, {165,10,10,10}, {166,1,9,15}, {166,3,10,14}, {166,4,12,12}, {167,6,10,13}, {168,2,9,15}, {168,9,9,12}, {169,0,12,13}, {169,6,7,15}, {170,1,12,13}, {170,4,10,14}, {171,0,0,18}, {171,3,9,15}, {171,5,12,12}, {171,6,9,14}, {172,0,1,18}, {172,2,12,13}, {173,1,1,18}, {174,0,2,18}, {174,7,10,13}, {175,0,10,15}, {175,1,2,18}, {175,3,12,13}, {175,4,9,15}, {175,5,10,14}, {176,1,10,15}, {176,7,7,15}, {177,0,3,18}, {177,2,2,18}, {177,6,12,12}, {178,1,3,18}, {178,2,10,15}, {178,7,9,14}, {178,9,10,12}, {179,4,12,13}, {180,2,3,18}, {180,5,9,15}, {181,0,4,18}, {181,3,10,15}, {181,6,10,14}, {181,9,9,13}, {182,0,13,13}, {182,1,4,18}, {183,0,12,14}, {183,1,13,13}, {183,3,3,18}, {184,1,12,14}, {184,2,4,18}, {184,5,12,13}, {184,7,12,12}, {185,2,13,13}, {185,4,10,15}, {186,0,5,18}, {186,2,12,14}, {186,6,9,15}, {187,1,5,18}, {187,3,4,18}, {188,3,13,13}, {188,7,10,14}, {188,10,10,12}, {189,2,5,18}, {189,3,12,14}, {190,0,0,19}, {190,5,10,15}, {190,6,12,13}, {191,0,1,19}, {191,4,4,18}, {191,9,10,13}, {192,0,6,18}, {192,1,1,19}, {192,3,5,18}, {192,4,13,13}, {193,0,2,19}, {193,1,6,18}, {193,4,12,14}, {193,7,9,15}, {194,1,2,19}, {195,2,6,18}, {195,9,9,14}, {196,0,3,19}, {196,0,13,14}, {196,2,2,19}, {196,4,5,18}, {196,6,10,15}, {197,1,3,19}, {197,1,13,14}, {197,5,13,13}, {197,7,12,13}, {198,0,12,15}, {198,3,6,18}, {198,5,12,14}, {199,0,7,18}, {199,1,12,15}, {199,2,3,19}, {199,2,13,14}, {200,0,4,19}, {200,1,7,18}, {201,1,4,19}, {201,2,12,15}, {201,5,5,18}, {201,9,12,12}, {201,10,10,13}, {202,2,7,18}, {202,3,3,19}, {202,3,13,14}, {202,4,6,18}, {203,2,4,19}, {203,6,13,13}, {203,7,10,15}, {204,3,12,15}, {204,6,12,14}, {205,0,5,19}, {205,3,7,18}, {205,9,10,14}, {206,1,5,19}, {206,3,4,19}, {206,4,13,14}, {207,5,6,18}, {208,2,5,19}, {208,4,12,15}, {209,4,7,18}, {210,0,0,20}, {210,0,14,14}, {210,4,4,19}, {210,7,13,13}, {210,9,9,15}, {211,0,1,20}, {211,0,6,19}, {211,0,13,15}, {211,1,14,14}, {211,3,5,19}, {211,5,13,14}, {211,7,12,14}, {211,10,12,12}, {212,1,1,20}, {212,1,6,19}, {212,1,13,15}, {213,0,2,20}, {213,2,14,14}, {213,5,12,15}, {213,6,6,18}, {214,1,2,20}, {214,2,6,19}, {214,2,13,15}, {214,5,7,18}, {214,9,12,13}, {215,4,5,19}, {215,10,10,14}, {216,0,3,20}, {216,0,9,18}, {216,2,2,20}, {216,3,14,14}, {217,1,3,20}, {217,1,9,18}, {217,3,6,19}, {217,3,13,15}, {217,6,13,14}, {218,0,7,19}, {219,1,7,19}, {219,2,3,20}, {219,2,9,18}, {219,6,12,15}, {220,0,4,20}, {220,4,14,14}, {220,5,5,19}, {220,6,7,18}, {220,9,10,15}, {221,1,4,20}, {221,2,7,19}, {221,4,6,19}, {221,4,13,15}, {222,3,3,20}, {222,3,9,18}, {223,2,4,20}, {224,3,7,19}, {224,7,13,14}, {224,10,12,13}, {225,0,5,20}, {225,0,14,15}, {225,5,14,14}, {226,0,10,18}, {226,1,5,20}, {226,1,14,15}, {226,3,4,20}, {226,4,9,18}, {226,5,6,19}, {226,5,13,15}, {226,7,12,15}, {227,1,10,18}, {227,7,7,18}, {227,9,13,13}, {228,2,5,20}, {228,2,14,15}, {228,4,7,19}, {228,9,12,14}, {229,2,10,18}, {230,4,4,20}, {230,10,10,15}, {231,0,0,21}, {231,0,6,20}, {231,3,5,20}, {231,3,14,15}, {231,5,9,18}, {231,6,14,14}, {232,0,1,21}, {232,1,6,20}, {232,3,10,18}, {232,6,6,19}, {232,6,13,15}, {233,1,1,21}, {233,5,7,19}, {234,0,2,21}, {234,2,6,20}, {234,12,12,12}, {235,0,9,19}, {235,1,2,21}, {235,4,5,20}, {235,4,14,15}, {236,1,9,19}, {236,4,10,18}, {237,0,3,21}, {237,2,2,21}, {237,3,6,20}, {237,6,9,18}, {237,10,13,13}, {238,0,7,20}, {238,1,3,21}, {238,2,9,19}, {238,7,14,14}, {238,10,12,14}, {239,1,7,20}, {239,6,7,19}, {239,7,13,15}, {240,0,15,15}, {240,2,3,21}, {240,5,5,20}, {240,5,14,15}, {241,0,4,21}, {241,1,15,15}, {241,2,7,20}, {241,3,9,19}, {241,4,6,20}, {241,5,10,18}, {241,9,13,14}, {242,1,4,21}, {243,2,15,15}, {243,3,3,21}, {243,9,12,15}, {244,2,4,21}, {244,3,7,20}, {244,7,9,18}, {245,0,10,19}, {245,4,9,19}, {246,0,5,21}, {246,1,10,19}, {246,3,15,15}, {246,5,6,20}, {246,6,14,15}, {246,7,7,19}, {247,1,5,21}, {247,3,4,21}, {247,6,10,18}, {247,12,12,13}, {248,2,10,19}, {248,4,7,20}, {249,0,12,18}, {249,2,5,21}, {250,1,12,18}, {250,4,15,15}, {250,5,9,19}, {251,3,10,19}, {251,4,4,21}, {251,10,13,14}, {252,0,6,21}, {252,2,12,18}, {252,3,5,21}, {252,6,6,20}, {253,0,0,22}, {253,1,6,21}, {253,5,7,20}, {253,7,14,15}, {253,10,12,15}, {254,0,1,22}, {254,7,10,18}, {255,0,9,20}, {255,1,1,22}, {255,2,6,21}, {255,3,12,18}, {255,4,10,19}, {255,5,15,15}, {255,9,14,14}, {256,0,2,22}, {256,1,9,20}, {256,4,5,21}, {256,6,9,19}, {256,9,13,15}, {257,1,2,22}, {258,2,9,20}, {258,3,6,21}, {259,0,3,22}, {259,0,7,21}, {259,2,2,22}, {259,4,12,18}, {259,6,7,20}, {260,1,3,22}, {260,1,7,21}, {260,5,10,19}, {260,12,13,13}, {261,3,9,20}, {261,5,5,21}, {261,6,15,15}, {261,9,9,18}, {261,12,12,14}, {262,0,13,18}, {262,2,3,22}, {262,2,7,21}, {262,4,6,21}, {263,0,4,22}, {263,1,13,18}, {263,7,9,19}, {264,1,4,22}, 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