A256313
Number of partitions of 3n into exactly 4 parts.
Original entry on oeis.org
0, 0, 2, 6, 15, 27, 47, 72, 108, 150, 206, 270, 351, 441, 551, 672, 816, 972, 1154, 1350, 1575, 1815, 2087, 2376, 2700, 3042, 3422, 3822, 4263, 4725, 5231, 5760, 6336, 6936, 7586, 8262, 8991, 9747, 10559, 11400, 12300, 13230, 14222, 15246, 16335, 17457
Offset: 0
For n=3 the 6 partitions of 3*3 = 9 are [1,1,1,6], [1,1,2,5], [1,1,3,4], [1,2,2,4], [1,2,3,3] and [2,2,2,3].
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).
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LinearRecurrence[{2,0,-2,2,-2,0,2,-1},{0,0,2,6,15,27,47,72},60] (* Harvey P. Dale, Jul 18 2021 *)
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concat(0, vector(40, n, k=0; forpart(p=3*n, k++, , [4,4]); k))
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concat([0,0], Vec(x^2*(x^2+2)*(x^2+x+1)/((x-1)^4*(x+1)^2*(x^2+1)) + O(x^100)))
A256314
Number of partitions of 3n into exactly 5 parts.
Original entry on oeis.org
0, 0, 1, 5, 13, 30, 57, 101, 164, 255, 377, 540, 748, 1014, 1342, 1747, 2233, 2818, 3507, 4319, 5260, 6351, 7599, 9027, 10642, 12470, 14518, 16814, 19366, 22204, 25337, 28796, 32591, 36756, 41301, 46262, 51649, 57501, 63829, 70673, 78045, 85987, 94512
Offset: 0
For n=3 the 5 partitions of 3*3 = 9 are [1,1,1,1,5], [1,1,1,2,4], [1,1,1,3,3], [1,1,2,2,3] and [1,2,2,2,2].
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-1,-2,2,1,-2,2,0,-2,1).
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Table[Length[IntegerPartitions[3n,{5}]],{n,0,50}] (* Harvey P. Dale, Jul 21 2019 *)
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concat(0, vector(40, n, k=0; forpart(p=3*n, k++, , [5,5]); k))
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concat([0,0], Vec(-x^2*(2*x^7+3*x^6+4*x^5+5*x^4+6*x^3+3*x^2+3*x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^4+x^3+x^2+x+1)) + O(x^100)))
A256524
Number of partitions of 3n into at most 4 parts.
Original entry on oeis.org
1, 3, 9, 18, 34, 54, 84, 120, 169, 225, 297, 378, 478, 588, 720, 864, 1033, 1215, 1425, 1650, 1906, 2178, 2484, 2808, 3169, 3549, 3969, 4410, 4894, 5400, 5952, 6528, 7153, 7803, 8505, 9234, 10018, 10830, 11700, 12600, 13561, 14553, 15609, 16698, 17854, 19044
Offset: 0
For n=1 the 3 partitions of 1*3 = 3 are [3], [1,2] and [1,1,1].
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).
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LinearRecurrence[{2, 0, -2, 2, -2, 0, 2, -1}, {1, 3, 9, 18, 34, 54, 84, 120}, 50] (* Jean-François Alcover, Apr 26 2017 *)
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concat(1, vector(40, n, k=0; forpart(p=3*n, k++, , [1,4]); k))
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Vec((x^2+x+1)*(2*x^2+1)/((x-1)^4*(x+1)^2*(x^2+1)) + O(x^100))
A256525
Number of partitions of 3n into at most 5 parts.
Original entry on oeis.org
1, 3, 10, 23, 47, 84, 141, 221, 333, 480, 674, 918, 1226, 1602, 2062, 2611, 3266, 4033, 4932, 5969, 7166, 8529, 10083, 11835, 13811, 16019, 18487, 21224, 24260, 27604, 31289, 35324, 39744, 44559, 49806, 55496, 61667, 68331, 75529, 83273, 91606, 100540
Offset: 0
For n=1 the 3 partitions of 1*3 = 3 are [3], [1,2] and [1,1,1].
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-1,-2,2,1,-2,2,0,-2,1).
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Table[Length[IntegerPartitions[3n,5]],{n,0,50}] (* Harvey P. Dale, Mar 08 2019 *)
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concat(1, vector(40, n, k=0; forpart(p=3*n, k++, , [1,5]); k))
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Vec(-(x^8+x^7+4*x^6+5*x^5+5*x^4+5*x^3+4*x^2+x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^4+x^3+x^2+x+1)) + O(x^100))
Showing 1-4 of 4 results.