|
| |
|
|
A309686
|
|
Sum of the even parts appearing among the smallest parts of the partitions of n into 3 parts.
|
|
11
|
|
|
|
0, 0, 0, 0, 0, 0, 2, 2, 4, 4, 6, 6, 12, 12, 18, 18, 24, 24, 36, 36, 48, 48, 60, 60, 80, 80, 100, 100, 120, 120, 150, 150, 180, 180, 210, 210, 252, 252, 294, 294, 336, 336, 392, 392, 448, 448, 504, 504, 576, 576, 648, 648, 720, 720, 810, 810, 900, 900, 990
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,7
|
|
|
LINKS
|
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,2,-2,-2,2,0,0,-1,1,1,-1).
|
|
|
FORMULA
|
a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} j * ((j-1) mod 2).
G.f.: 2*x^6 / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x + x^2)^2).
a(n) = a(n-1) + a(n-2) - a(n-3) + 2*a(n-6) - 2*a(n-7) - 2*a(n-8) + 2*a(n-9) - a(n-12) + a(n-13) + a(n-14) - a(n-15) for n>14.
(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) | 0 0 0 2 2 4 4 6 ...
-----------------------------------------------------------------------
|
|
|
MATHEMATICA
|
Table[Sum[Sum[j*Mod[j - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
LinearRecurrence[{1, 1, -1, 0, 0, 2, -2, -2, 2, 0, 0, -1, 1, 1, -1}, {0, 0, 0, 0, 0, 0, 2, 2, 4, 4, 6, 6, 12, 12, 18}, 80]
|
|
|
CROSSREFS
|
Cf. A026923, A026927, A309683, A309684, A309685, A309687, A309688, A309689, A309690, A309692, A309694.
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
