Derive a formula for the number of partitions of n into parts that are odd and bigger than 1; e.g. a(12)=5 cases: 3+3+3+3, 5+7, 7+5, 3+9, 9+3.

Verify your formula by evaluating a(14).

DECLARE SUB decompose (x!)
CLEAR , , 25000
DIM SHARED ct
CLS
FOR n = 1 TO 44
  ct = 0
  decompose n
  PRINT n, ct
NEXT n

SUB decompose (x)
  IF x < 3 THEN EXIT SUB
  FOR addend = 3 TO x STEP 2
   IF addend = x THEN
    ct = ct + 1
   ELSE
    decompose x - addend
   END IF
  NEXT
END SUB

finds

n         partitions   
1             0
2             0
3             1
4             0
5             1
6             1
7             1
8             2
9             2
10            3
11            4
12            5
13            7
14            9
15            12
16            16
17            21
18            28
19            37
20            49
21            65
22            86
23            114
24            151
25            200
26            265
27            351
28            465
29            616
30            816
31            1081
32            1432
33            1897
34            2513
35            3329
36            4410
37            5842
38            7739
39            10252
40            13581
41            17991
42            23833
43            31572
44            41824

Posted by Charlie on 2013-01-24 13:55:44