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%I A351010 #12 Feb 06 2022 23:11:02
%S A351010 0,3,10,15,36,43,58,63,136,147,170,175,228,235,250,255,528,547,586,
%T A351010 591,676,683,698,703,904,915,938,943,996,1003,1018,1023,2080,2115,
%U A351010 2186,2191,2340,2347,2362,2367,2696,2707,2730,2735,2788,2795,2810,2815,3600,3619
%N A351010 Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).
%C A351010 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 A351010 The terms together with their binary expansions and the corresponding compositions begin:
%e A351010     0:         0  ()
%e A351010     3:        11  (1,1)
%e A351010    10:      1010  (2,2)
%e A351010    15:      1111  (1,1,1,1)
%e A351010    36:    100100  (3,3)
%e A351010    43:    101011  (2,2,1,1)
%e A351010    58:    111010  (1,1,2,2)
%e A351010    63:    111111  (1,1,1,1,1,1)
%e A351010   136:  10001000  (4,4)
%e A351010   147:  10010011  (3,3,1,1)
%e A351010   170:  10101010  (2,2,2,2)
%e A351010   175:  10101111  (2,2,1,1,1,1)
%e A351010   228:  11100100  (1,1,3,3)
%e A351010   235:  11101011  (1,1,2,2,1,1)
%e A351010   250:  11111010  (1,1,1,1,2,2)
%e A351010   255:  11111111  (1,1,1,1,1,1,1,1)
%t A351010 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A351010 Select[Range[0,100],And@@EvenQ/@Length/@Split[stc[#]]&]
%Y A351010 The case of twins (binary weight 2) is A000120.
%Y A351010 The Heinz numbers of these compositions are given by A000290.
%Y A351010 All terms are evil numbers A001969.
%Y A351010 Partitions of this type are counted by A035363, any length A351004.
%Y A351010 These compositions are counted by A077957(n-2), see also A016116.
%Y A351010 The strict case (distinct twins) is A351009, counted by A032020 with 0's.
%Y A351010 The anti-run case is A351011, counted by A003242 interspersed with 0's.
%Y A351010 A011782 counts integer compositions.
%Y A351010 A085207/A085208 represent concatenation of standard compositions.
%Y A351010 A333489 ranks anti-runs, complement A348612.
%Y A351010 A345167/A350355/A350356 rank alternating compositions.
%Y A351010 A351014 counts distinct runs in standard compositions.
%Y A351010 Cf. A018819, A025047, A027383, A035457, A053738, A088218, A106356, A238279, A344604, A351012, A351015.
%Y A351010 Selected statistics of standard compositions:
%Y A351010 - Length is A000120.
%Y A351010 - Sum is A070939.
%Y A351010 - Heinz number is A333219.
%Y A351010 - Number of distinct parts is A334028.
%Y A351010 Selected classes of standard compositions:
%Y A351010 - Partitions are A114994, strict A333256.
%Y A351010 - Multisets are A225620, strict A333255.
%Y A351010 - Strict compositions are A233564.
%Y A351010 - Constant compositions are A272919.
%K A351010 nonn
%O A351010 1,2
%A A351010 _Gus Wiseman_, Feb 01 2022

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