|
| |
|
|
A308266
|
|
Sum of the middle parts in the partitions of n into 3 parts.
|
|
1
|
|
|
|
0, 0, 1, 1, 3, 5, 8, 11, 18, 22, 31, 40, 51, 62, 80, 93, 114, 135, 159, 183, 217, 244, 282, 320, 362, 404, 459, 505, 565, 625, 690, 755, 836, 906, 993, 1080, 1173, 1266, 1378, 1477, 1596, 1715, 1841, 1967, 2115, 2248, 2404, 2560, 2724, 2888, 3077, 3249, 3447
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,5
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} i.
G.f.: x^3*(1 + x + x^2 + x^3 + x^4) / ((1 - x)^4*(1 + x)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) - 4*a(n-5) - a(n-6) + 2*a(n-7) + 2*a(n-8) - a(n-10) for n>10.
(End)
|
|
|
EXAMPLE
|
Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
1+1+8
1+1+7 1+2+7
1+2+6 1+3+6
1+1+6 1+3+5 1+4+5
1+1+5 1+2+5 1+4+4 2+2+6
1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
-----------------------------------------------------------------------
n | 3 4 5 6 7 8 9 10 ...
-----------------------------------------------------------------------
a(n) | 1 1 3 5 8 11 18 22 ...
-----------------------------------------------------------------------
|
|
|
MATHEMATICA
|
Table[Sum[Sum[i, {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
Table[Total[IntegerPartitions[n, {3}][[All, 2]]], {n, 60}] (* Harvey P. Dale, Jul 09 2020 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
