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Irregular triangle read by rows where row n lists the positions of zeros in the reversed binary expansion of n.
5

%I #7 May 29 2024 07:04:50

%S 1,1,2,2,1,1,2,3,2,3,1,3,3,1,2,2,1,1,2,3,4,2,3,4,1,3,4,3,4,1,2,4,2,4,

%T 1,4,4,1,2,3,2,3,1,3,3,1,2,2,1,1,2,3,4,5,2,3,4,5,1,3,4,5,3,4,5,1,2,4,

%U 5,2,4,5,1,4,5,4,5,1,2,3,5,2,3,5,1,3,5

%N Irregular triangle read by rows where row n lists the positions of zeros in the reversed binary expansion of n.

%e The reversed binary expansion of 100 is (0,0,1,0,0,1,1), with zeros at positions {1,2,4,5}, so row 100 is (1,2,4,5).

%e Triangle begins:

%e 1:

%e 2: 1

%e 3:

%e 4: 1 2

%e 5: 2

%e 6: 1

%e 7:

%e 8: 1 2 3

%e 9: 2 3

%e 10: 1 3

%e 11: 3

%e 12: 1 2

%e 13: 2

%e 14: 1

%e 15:

%e 16: 1 2 3 4

%t Table[Join@@Position[Reverse[IntegerDigits[n,2]],0],{n,30}]

%Y Row lengths are A023416, partial sums A059015.

%Y For ones instead of zeros we have A048793, lengths A000120, sums A029931.

%Y Row sums are A359400, non-reversed A359359.

%Y Same as A368494 but with empty rows () instead of (0).

%Y A003714 lists numbers with no successive binary indices.

%Y A030190 gives binary expansion, reverse A030308.

%Y A039004 lists the positions of zeros in A345927.

%Y Cf. A069010, A070939, A073642, A230877, A344618, A359402, A359495.

%K nonn,base,tabf

%O 1,3

%A _Gus Wiseman_, May 26 2024