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A351009 Numbers k such that the k-th composition in standard order is a concatenation of distinct twins (x,x).

%I A351009 #12 Mar 13 2022 19:00:03
%S A351009 0,3,10,36,43,58,136,147,228,528,547,586,676,904,2080,2115,2186,2347,
%T A351009 2362,2696,2707,2788,3600,3658,3748,8256,8323,8458,8740,8747,8762,
%U A351009 9352,10768,10787,11144,14368,14474,14984,32896,33027,33290,33828,33835,33850,34963
%N A351009 Numbers k such that the k-th composition in standard order is a concatenation of distinct twins (x,x).
%C A351009 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 A351009 The terms together with their binary expansions and standard compositions begin:
%e A351009     0:           0  ()
%e A351009     3:          11  (1,1)
%e A351009    10:        1010  (2,2)
%e A351009    36:      100100  (3,3)
%e A351009    43:      101011  (2,2,1,1)
%e A351009    58:      111010  (1,1,2,2)
%e A351009   136:    10001000  (4,4)
%e A351009   147:    10010011  (3,3,1,1)
%e A351009   228:    11100100  (1,1,3,3)
%e A351009   528:  1000010000  (5,5)
%e A351009   547:  1000100011  (4,4,1,1)
%e A351009   586:  1001001010  (3,3,2,2)
%e A351009   676:  1010100100  (2,2,3,3)
%e A351009   904:  1110001000  (1,1,4,4)
%t A351009 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]], 1],0]]//Reverse;
%t A351009 Select[Range[0,1000], UnsameQ@@Split[stc[#]]&&And@@(#==2&)/@Length/@Split[stc[#]]&]
%Y A351009 The case of twins (binary weight 2) is A000120.
%Y A351009 All terms are evil numbers A001969.
%Y A351009 The version for Heinz numbers of partitions is A062503, counted by A035457.
%Y A351009 These compositions are counted by A032020 interspersed with 0's.
%Y A351009 Taking singles instead of twins gives A349051.
%Y A351009 This is the strict (distinct twins) version of A351010 and A351011.
%Y A351009 A011782 counts compositions.
%Y A351009 A085207 represents concatenation using standard compositions.
%Y A351009 A333489 ranks anti-runs, complement A348612.
%Y A351009 A345167 ranks alternating compositions, counted by A025047.
%Y A351009 A351014 counts distinct runs in standard compositions, see A351015.
%Y A351009 Cf. A003242, A027383, A035363, A088218, A106356, A122134, A238279, A344604, A349054, A351005, A351007.
%Y A351009 Selected statistics of standard compositions:
%Y A351009 - Length is A000120.
%Y A351009 - Sum is A070939.
%Y A351009 - Heinz number is A333219.
%Y A351009 - Number of distinct parts is A334028.
%Y A351009 Selected classes of standard compositions:
%Y A351009 - Partitions are A114994, strict A333256.
%Y A351009 - Multisets are A225620, strict A333255.
%Y A351009 - Strict compositions are A233564.
%Y A351009 - Constant compositions are A272919.
%K A351009 nonn
%O A351009 1,2
%A A351009 _Gus Wiseman_, Feb 03 2022