|
|
A227568
|
|
Largest k such that a partition of n into distinct parts with boundary size k exists.
|
|
3
|
|
|
0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
The boundary size is the number of parts having fewer than two neighbors.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = floor(2*sqrt(n/3)).
|
|
MAPLE
|
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, 1, 0),
`if`(i<1, 0, max(`if`(t>1, 1, 0)+b(n, i-1, iquo(t, 2)),
`if`(i>n, 0, `if`(t=2, 1, 0)+b(n-i, i-1, iquo(t, 2)+2)))))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);
|
|
MATHEMATICA
|
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t > 1, 1, 0], If[i < 1, 0, Max[If[t > 1, 1, 0] + b[n, i - 1, Quotient[t, 2]], If[i > n, 0, If[t == 2, 1, 0] + b[n - i, i - 1, Quotient[t, 2] + 2]]]]];
a[n_] := b[n, n, 0];
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|