|
| |
|
|
A182714
|
|
Number of 4's in the outer shell of the set of partitions of n.
|
|
18
|
|
|
|
0, 0, 0, 1, 0, 1, 1, 3, 2, 5, 5, 10, 10, 17, 19, 31, 34, 51, 60, 86, 100, 139, 165, 223, 265, 349, 418, 543, 648, 827, 992, 1251, 1495, 1866, 2230, 2758, 3289, 4033, 4803, 5852, 6949, 8411, 9973, 12005, 14194, 17002, 20060, 23919, 28153, 33426, 39256, 46438
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,8
|
|
|
COMMENTS
|
Zero together with the first differences of A024788. Also number of 4's in all partitions of n that do not contain 1 as a part.
|
|
|
LINKS
|
Alois P. Heinz, Table of n, a(n) for n = 1..1000
|
|
|
FORMULA
|
It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3) + a(n+4), n >= 0. - Omar E. Pol, Feb 04 2012
|
|
|
EXAMPLE
|
a(8) = 3 counts the 4's in 8 = 4+4 = 4+2+2. The 4's in 8 = 4+3+1 = 4+2+1+1 = 4+1+1+1+1 do not count.
Contribution from Omar E. Pol, _Oct 25 2012_ (Start):
--------------------------------------
. Number
Outer shell of 8 of 4's
--------------------------------------
8 .............................. 0
4 + 4 .......................... 2
5 + 3 .......................... 0
6 + 2 .......................... 0
3 + 3 + 2 ...................... 0
4 + 2 + 2 ...................... 1
2 + 2 + 2 + 2 .................. 0
. 1 .......................... 0
. 1 ...................... 0
. 1 ...................... 0
. 1 .................. 0
. 1 ...................... 0
. 1 .................. 0
. 1 .................. 0
. 1 .............. 0
. 1 .................. 0
. 1 .............. 0
. 1 .............. 0
. 1 .......... 0
. 1 .......... 0
. 1 ...... 0
. 1 .. 0
------------------------------------
. 6 - 3 = 3
The difference between the sum of the fourth column and the sum of the fifth column of the outer shell of 8 is 6 - 3 = 3 and equals the number of 4's in the outer shell of the set of partitions of 8, so a(8) = 3 (see also A024788).
(End)
|
|
|
MAPLE
|
b:= proc(n, i) option remember; local g, h;
if n=0 then [1, 0]
elif i<2 then [0, 0]
else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
[g[1]+h[1], g[2]+h[2]+`if`(i=4, h[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..70); # Alois P. Heinz, Mar 19 2012
|
|
|
PROG
|
(Sage) A182714 = lambda n: sum(list(p).count(4) for p in Partitions(n) if 1 not in p)
|
|
|
CROSSREFS
|
Column 4 of A194812.
Cf. A015739, A024788, A135010, A138121, A182703, A182712, A182713.
Sequence in context: A186545 A008623 A035546 * A198755 A134237 A099889
Adjacent sequences: A182711 A182712 A182713 * A182715 A182716 A182717
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Omar E. Pol, Nov 13 2011
|
|
|
STATUS
|
approved
|
| |
|
|
