A329143 - OEIS

%I #10 Jun 27 2020 03:43:28

%S 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,

%T 20,11,18,12,27,8,27,16,32,14,35,14,42,23,43,17,56,17,61,31,67,25,78,

%U 28,88,41,89,35,119,39,116,60,131,52,154,52,170,75,182

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

%C 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).

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

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

%e The a(n) partitions for n = 2, 5, 8, 14, 16, 22:

%e   11  32     53        95              5533              7744

%e       11111  3221      5432            7441              9652

%e              11111111  32222111        533311            554332

%e                        11111111111111  33222211          54333211

%e                                        1111111111111111  332222221111

%e                                                          1111111111111111111111

%t aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];

%t aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];

%t Table[Length[Select[IntegerPartitions[n],!aperQ[aug[#]]&]],{n,0,30}]

%Y The Heinz numbers of these partitions are given by A329132.

%Y The aperiodic version is A329136.

%Y The non-augmented version is A329144.

%Y Periodic binary words are A152061.

%Y Periodic compositions are A178472.

%Y Numbers whose binary expansion is periodic are A121016.

%Y Numbers whose prime signature is periodic are A329140.

%Y Cf. A000740, A027375, A328594, A329133, A329135, A325351, A325356, A329134.

%K nonn

%O 0,6

%A _Gus Wiseman_, Nov 10 2019

%E More terms from _Jinyuan Wang_, Jun 27 2020