A328599 Number of compositions of n with no part circularly followed by a divisor or a multiple.

1, 0, 0, 0, 0, 2, 0, 4, 2, 4, 12, 8, 22, 14, 36, 44, 62, 114, 130, 206, 264, 414, 602, 822, 1250, 1672, 2520, 3518, 5146, 7408, 10448, 15224, 21496, 31284, 44718, 64170, 92314, 131618, 190084, 271870, 391188, 560978, 804264, 1155976, 1656428, 2381306, 3414846

Offset: 0

Author

Gus Wiseman, Oct 25 2019

Keywords

• nonn

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.

Examples

			The a(0) = 1 through a(12) = 22 compositions (empty columns 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)
                                  (2,3,5)    (6,5)  (3,2,7)
                                  (2,5,3)    (7,4)  (3,4,5)
                                  (3,2,5)    (8,3)  (3,5,4)
                                  (3,5,2)    (9,2)  (3,7,2)
                                  (5,2,3)           (4,3,5)
                                  (5,3,2)           (4,5,3)
                                  (2,3,2,3)         (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)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

The necklace version is A328601.

The case forbidding only divisors (not multiples) is A328598.

The non-circular version is A328508.

Partitions with no part followed by a divisor are A328171.

Cf. A000740, A008965, A167606, A318729, A318748, A328460, A328593, A328600, A328603, A328608, A328609, A328674.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Not/@Divisible@@@Partition[#,2,1,1]&&And@@Not/@Divisible@@@Reverse/@Partition[#,2,1,1]&]],{n,0,10}]

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&&j%i<>0)))} \\ Andrew Howroyd, Oct 26 2019

Extensions

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019