A329144 Number of integer partitions of n whose differences are a periodic word.

0, 0, 1, 1, 1, 3, 1, 2, 5, 3, 2, 8, 2, 5, 9, 7, 5, 12, 7, 7, 19, 9, 9, 21, 12, 15, 23, 18, 17, 29, 21, 19, 42, 23, 31, 42, 38, 29, 53, 43, 44, 62, 49, 52, 79, 55, 72, 75, 87, 63, 117, 79, 104, 107, 120, 99, 156, 117, 143, 147

Offset: 1

Author

Gus Wiseman, Nov 10 2019

Keywords

• nonn

Comments

A finite sequence is periodic if its cyclic rotations are not all different.

Examples

			The a(n) partitions for n = 3, 6, 8, 9, 12, 15, 16:
  111  222     2222      333        444           555              4444
       321     11111111  432        543           654              7531
       111111            531        642           753              44332
                         32211      741           852              3332221
                         111111111  3333          951              4332211
                                    222222        33333            22222222
                                    3222111       54321            1111111111111111
                                    111111111111  322221111
                                                  111111111111111
		

Crossrefs

The Heinz numbers of these partitions are given by A329134.

The augmented version is A329143.

Periodic binary words are A152061.

Periodic compositions are A178472.

Numbers whose binary expansion is periodic are A121016.

Numbers whose prime signature is periodic are A329140.

Cf. A000740, A027375, A059966, A328594, A328596, A329132, A329135, A329136.

Programs

Mathematica

aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Table[Length[Select[IntegerPartitions[n],!aperQ[Differences[#]]&]],{n,30}]