A302392 Number of odd parts in the partitions of 3n into 3 parts.

3, 4, 13, 18, 33, 40, 61, 72, 99, 112, 145, 162, 201, 220, 265, 288, 339, 364, 421, 450, 513, 544, 613, 648, 723, 760, 841, 882, 969, 1012, 1105, 1152, 1251, 1300, 1405, 1458, 1569, 1624, 1741, 1800, 1923, 1984, 2113, 2178, 2313, 2380, 2521, 2592, 2739, 2812

Offset

1,1

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*(3 + x + 6*x^2 + 4*x^3 + 3*x^4 + x^5) / ((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 odd 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
---------------------------------------------------------------------
    3           4           13          18          33      ..  a(n)

Mathematica

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

Crossrefs

Cf. A302393 (even parts).

Sequence in context: A323149 A291250 A205901 * A293941 A182691 A358583

Adjacent sequences: A302389 A302390 A302391 * A302393 A302394 A302395

Keyword

nonn

Author

Wesley Ivan Hurt, Apr 07 2018

Status

approved