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A329143
Number of integer partitions of n whose augmented differences are a periodic word.
4
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.
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.
Sequence in context: A326036 A133776 A060118 * A219032 A234567 A241950
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 10 2019
EXTENSIONS
More terms from Jinyuan Wang, Jun 27 2020
STATUS
approved