%I #12 Feb 06 2022 23:11:02

%S 0,3,10,15,36,43,58,63,136,147,170,175,228,235,250,255,528,547,586,

%T 591,676,683,698,703,904,915,938,943,996,1003,1018,1023,2080,2115,

%U 2186,2191,2340,2347,2362,2367,2696,2707,2730,2735,2788,2795,2810,2815,3600,3619

%N Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).

%C The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

%e The terms together with their binary expansions and the corresponding compositions begin:

%e 0: 0 ()

%e 3: 11 (1,1)

%e 10: 1010 (2,2)

%e 15: 1111 (1,1,1,1)

%e 36: 100100 (3,3)

%e 43: 101011 (2,2,1,1)

%e 58: 111010 (1,1,2,2)

%e 63: 111111 (1,1,1,1,1,1)

%e 136: 10001000 (4,4)

%e 147: 10010011 (3,3,1,1)

%e 170: 10101010 (2,2,2,2)

%e 175: 10101111 (2,2,1,1,1,1)

%e 228: 11100100 (1,1,3,3)

%e 235: 11101011 (1,1,2,2,1,1)

%e 250: 11111010 (1,1,1,1,2,2)

%e 255: 11111111 (1,1,1,1,1,1,1,1)

%t stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[0,100],And@@EvenQ/@Length/@Split[stc[#]]&]

%Y The case of twins (binary weight 2) is A000120.

%Y The Heinz numbers of these compositions are given by A000290.

%Y All terms are evil numbers A001969.

%Y Partitions of this type are counted by A035363, any length A351004.

%Y These compositions are counted by A077957(n-2), see also A016116.

%Y The strict case (distinct twins) is A351009, counted by A032020 with 0's.

%Y The anti-run case is A351011, counted by A003242 interspersed with 0's.

%Y A011782 counts integer compositions.

%Y A085207/A085208 represent concatenation of standard compositions.

%Y A333489 ranks anti-runs, complement A348612.

%Y A345167/A350355/A350356 rank alternating compositions.

%Y A351014 counts distinct runs in standard compositions.

%Y Cf. A018819, A025047, A027383, A035457, A053738, A088218, A106356, A238279, A344604, A351012, A351015.

%Y Selected statistics of standard compositions:

%Y - Length is A000120.

%Y - Sum is A070939.

%Y - Heinz number is A333219.

%Y - Number of distinct parts is A334028.

%Y Selected classes of standard compositions:

%Y - Partitions are A114994, strict A333256.

%Y - Multisets are A225620, strict A333255.

%Y - Strict compositions are A233564.

%Y - Constant compositions are A272919.

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 01 2022