A302393 Number of even parts in the partitions of 3n into 3 parts.

0, 5, 8, 18, 24, 41, 50, 72, 84, 113, 128, 162, 180, 221, 242, 288, 312, 365, 392, 450, 480, 545, 578, 648, 684, 761, 800, 882, 924, 1013, 1058, 1152, 1200, 1301, 1352, 1458, 1512, 1625, 1682, 1800, 1860, 1985, 2048, 2178, 2244, 2381, 2450, 2592, 2664, 2813

Offset

1,2

Links

Table of n, a(n) for n=1..50.

Index entries for sequences related to partitions

Formula

Conjectures from Colin Barker, Apr 07 2018: (Start)

G.f.: x^2*(5 + 3*x + 5*x^2 + 3*x^3 + 2*x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)).

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7) for n>7.

(End)

Example

Count the even parts for a(n) (n > 0).
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
---------------------------------------------------------------------
    0           5           8           18          24      ..  a(n)

Mathematica

Table[Count[Flatten[IntegerPartitions[3 n, {3}]], _?EvenQ], {n, 100}]

Crossrefs

Cf. A302392 (odd parts).

Sequence in context: A155086 A219049 A245534 * A342804 A226902 A347136

Adjacent sequences: A302390 A302391 A302392 * A302394 A302395 A302396

Keyword

nonn

Author

Wesley Ivan Hurt, Apr 07 2018

Status

approved