|
|
A328598
|
|
Number of compositions of n with no part circularly followed by a divisor.
|
|
12
|
|
|
1, 0, 0, 0, 0, 2, 0, 4, 2, 7, 12, 11, 22, 26, 55, 63, 99, 149, 215, 324, 458, 699, 1006, 1492, 2185, 3202, 4734, 6928, 10242, 14951, 22023, 32365, 47557, 69905, 102633, 150983, 221712, 325918, 478841, 703647, 1034103, 1519431, 2233061, 3281003, 4821790, 7085358
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,6
|
|
COMMENTS
|
A composition of n is a finite sequence of positive integers summing to n.
Circularity means the last part is followed by the first.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
The a(5) = 2 through a(12) = 22 compositions (empty column not shown):
(2,3) (2,5) (3,5) (2,7) (3,7) (2,9) (5,7)
(3,2) (3,4) (5,3) (4,5) (4,6) (3,8) (7,5)
(4,3) (5,4) (6,4) (4,7) (2,3,7)
(5,2) (7,2) (7,3) (5,6) (2,7,3)
(2,4,3) (2,3,5) (6,5) (3,2,7)
(3,2,4) (2,5,3) (7,4) (3,4,5)
(4,3,2) (3,2,5) (8,3) (3,5,4)
(3,5,2) (9,2) (3,7,2)
(5,2,3) (2,4,5) (4,3,5)
(5,3,2) (4,5,2) (4,5,3)
(2,3,2,3) (5,2,4) (5,3,4)
(3,2,3,2) (5,4,3)
(7,2,3)
(7,3,2)
(2,3,2,5)
(2,3,4,3)
(2,5,2,3)
(3,2,3,4)
(3,2,5,2)
(3,4,3,2)
(4,3,2,3)
(5,2,3,2)
|
|
MATHEMATICA
|
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@Not/@Divisible@@@Partition[#, 2, 1, 1]&]], {n, 0, 10}]
|
|
PROG
|
(PARI)
b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={concat([1], sum(k=1, n, b(n, k, (i, j)->i%j<>0)))} \\ Andrew Howroyd, Oct 26 2019
|
|
CROSSREFS
|
The version with singletons is A318726.
The non-circular version is A328460.
Also forbidding parts circularly followed by a multiple gives A328599.
Partitions with no part followed by a divisor are A328171.
Cf. A000740, A008965, A167606, A178470, A318748, A328187, A328508, A328593, A328597, A328601, A328603, A328609.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|