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%I #8 Nov 14 2019 12:58:06
%S 1,1,1,2,4,5,10,14,19,28,40,53,75,99,131,172,226,294,380,488,617,787,
%T 996,1250,1565,1953,2425,3003,3705,4559,5589,6836,8329,10132,12292,
%U 14871,17950,21629,25988,31169,37306,44569,53139,63247,75133,89111,105515,124737
%N Number of integer partitions of n whose augmented differences are an aperiodic 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 sequence is aperiodic if its cyclic rotations are all different.
%F a(n) + A329143(n) = A000041(n).
%e The a(1) = 1 through a(7) = 14 partitions:
%e (1) (2) (3) (4) (5) (6) (7)
%e (2,1) (2,2) (4,1) (3,3) (4,3)
%e (3,1) (2,2,1) (4,2) (5,2)
%e (2,1,1) (3,1,1) (5,1) (6,1)
%e (2,1,1,1) (2,2,2) (3,2,2)
%e (3,2,1) (3,3,1)
%e (4,1,1) (4,2,1)
%e (2,2,1,1) (5,1,1)
%e (3,1,1,1) (2,2,2,1)
%e (2,1,1,1,1) (3,2,1,1)
%e (4,1,1,1)
%e (2,2,1,1,1)
%e (3,1,1,1,1)
%e (2,1,1,1,1,1)
%e With augmented differences:
%e (1) (2) (3) (4) (5) (6) (7)
%e (2,1) (1,2) (4,1) (1,3) (2,3)
%e (3,1) (1,2,1) (3,2) (4,2)
%e (2,1,1) (3,1,1) (5,1) (6,1)
%e (2,1,1,1) (1,1,2) (1,3,1)
%e (2,2,1) (2,1,2)
%e (4,1,1) (3,2,1)
%e (1,2,1,1) (5,1,1)
%e (3,1,1,1) (1,1,2,1)
%e (2,1,1,1,1) (2,2,1,1)
%e (4,1,1,1)
%e (1,2,1,1,1)
%e (3,1,1,1,1)
%e (2,1,1,1,1,1)
%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 A329133.
%Y The periodic version is A329143.
%Y The non-augmented version is A329137.
%Y Aperiodic binary words are A027375.
%Y Aperiodic compositions are A000740.
%Y Numbers whose binary expansion is aperiodic are A328594.
%Y Numbers whose differences of prime indices are aperiodic are A329135.
%Y Numbers whose prime signature is aperiodic are A329139.
%Y Cf. A152061, A325351, A325356, A329132, A329134, A329139, A329140.
%K nonn
%O 0,4
%A _Gus Wiseman_, Nov 09 2019
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