A328609 Number of compositions of n whose circularly adjacent parts are relatively prime.

1, 1, 1, 3, 6, 12, 23, 42, 81, 150, 284, 534, 1004, 1882, 3532, 6630, 12459, 23406, 43951, 82537, 154998, 291087, 546673, 1026686, 1928117, 3621016, 6800299, 12771085, 23984328, 45042958, 84591338, 158863806, 298348612, 560303341, 1052258401, 1976157509

Offset

0,4

Comments

Circularity means the last part is followed by the first.

Links

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

Formula

a(n > 1) = A318748(n) - 1.

Example

The a(1) = 1 through a(6) = 23 compositions:
  (1)  (11)  (12)   (13)    (14)     (15)
             (21)   (31)    (23)     (51)
             (111)  (112)   (32)     (114)
                    (121)   (41)     (123)
                    (211)   (113)    (132)
                    (1111)  (131)    (141)
                            (311)    (213)
                            (1112)   (231)
                            (1121)   (312)
                            (1211)   (321)
                            (2111)   (411)
                            (11111)  (1113)
                                     (1131)
                                     (1212)
                                     (1311)
                                     (2121)
                                     (3111)
                                     (11112)
                                     (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@CoprimeQ@@@Partition[#, 2, 1, 1]&]], {n, 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)->gcd(i, j)==1)))} \\ Andrew Howroyd, Nov 01 2019

Crossrefs

The necklace version is A328597 or A318728 (with singletons).

The aperiodic version is A328670.

The Lyndon word version is A318745.

The version with singletons is A318748.

The non-circular version is A167606.

Relatively prime compositions are A000740.

Compositions with no part circularly followed by a divisor are A328598.

Cf. A008965, A178470, A318726, A325680, A328172, A328335, A328599, A328600.

Sequence in context: A327546 A181844 A162506 * A227681 A055244 A089068

Adjacent sequences: A328606 A328607 A328608 * A328610 A328611 A328612

Keyword

nonn

Author

Gus Wiseman, Oct 26 2019

Status

approved