|
| |
|
|
A308034
|
|
Number of partitions of n into 3 parts with at least 1 part that is a nondivisor of n.
|
|
0
|
|
|
|
0, 0, 0, 0, 2, 1, 4, 4, 6, 8, 10, 9, 14, 16, 18, 20, 24, 25, 30, 32, 36, 40, 44, 45, 52, 56, 60, 64, 70, 73, 80, 84, 90, 96, 102, 105, 114, 120, 126, 132, 140, 145, 154, 160, 168, 176, 184, 189, 200, 208, 216, 224, 234, 241, 252, 260, 270, 280, 290, 297, 310
(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)} 1 - chi(n/i) * chi(n/k) * chi(n/(n-i-k)), where chi(n) = 1 - ceiling(n) + floor(n).
G.f.: x^5*(2 - 3*x + 6*x^2 - 6*x^3 + 6*x^4 - 3*x^5) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) for n>10.
(End)
|
|
|
EXAMPLE
|
7 = 2 + (1 + 4) = 2 + (2 + 3) = 3 + (1 + 3) = 5 + (1 + 1); the first integer corresponds to one part that is a nondivisor of 7. So a(7) = 4. - Bernard Schott, May 12 2019
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 2 1 4 4 6 8 ...
-----------------------------------------------------------------------
|
|
|
MATHEMATICA
|
Table[Sum[Sum[1 - (1 - Ceiling[n/i] + Floor[n/i])*(1 - Ceiling[n/k] + Floor[n/k])*(1 - Ceiling[n/(n - i - k)] + Floor[n/(n - i - k)]), {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
