A329143 Number of integer partitions of n whose augmented differences are a periodic word.

0, 0, 1, 1, 1, 2, 1, 1, 3, 2, 2, 3, 2, 2, 4, 4, 5, 3, 5, 2, 10, 5, 6, 5, 10, 5, 11, 7, 13, 6, 15, 6, 20, 11, 18, 12, 27, 8, 27, 16, 32, 14, 35, 14, 42, 23, 43, 17, 56, 17, 61, 31, 67, 25, 78, 28, 88, 41, 89, 35, 119, 39, 116, 60, 131, 52, 154, 52, 170, 75, 182

Offset

0,6

Comments

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

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

Links

Table of n, a(n) for n=0..70.

Formula

a(n) + A329136(n) = A000041(n).

Example

The a(n) partitions for n = 2, 5, 8, 14, 16, 22:
  11  32     53        95              5533              7744
      11111  3221      5432            7441              9652
             11111111  32222111        533311            554332
                       11111111111111  33222211          54333211
                                       1111111111111111  332222221111
                                                         1111111111111111111111

Mathematica

aperQ[q_]:=Array[RotateRight[q, #1]&, Length[q], 1, UnsameQ];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], !aperQ[aug[#]]&]], {n, 0, 30}]

Crossrefs

The Heinz numbers of these partitions are given by A329132.

The aperiodic version is A329136.

The non-augmented version is A329144.

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, A328594, A329133, A329135, A325351, A325356, A329134.

Sequence in context: A326036 A133776 A060118 * A219032 A234567 A241950

Adjacent sequences: A329140 A329141 A329142 * A329144 A329145 A329146

Keyword

nonn

Author

Gus Wiseman, Nov 10 2019

Extensions

More terms from Jinyuan Wang, Jun 27 2020

Status

approved