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A351011 Numbers k such that the k-th composition in standard order has even length and alternately equal and unequal parts, i.e., all run-lengths equal to 2.

%I A351011 #9 Feb 06 2022 23:11:09
%S A351011 0,3,10,36,43,58,136,147,228,235,528,547,586,676,698,904,915,2080,
%T A351011 2115,2186,2347,2362,2696,2707,2788,2795,3600,3619,3658,3748,3770,
%U A351011 8256,8323,8458,8740,8747,8762,9352,9444,9451,10768,10787,10826,11144,11155,14368
%N A351011 Numbers k such that the k-th composition in standard order has even length and alternately equal and unequal parts, i.e., all run-lengths equal to 2.
%C A351011 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 A351011 The terms together with their binary expansions and standard compositions begin:
%e A351011     0:           0  ()
%e A351011     3:          11  (1,1)
%e A351011    10:        1010  (2,2)
%e A351011    36:      100100  (3,3)
%e A351011    43:      101011  (2,2,1,1)
%e A351011    58:      111010  (1,1,2,2)
%e A351011   136:    10001000  (4,4)
%e A351011   147:    10010011  (3,3,1,1)
%e A351011   228:    11100100  (1,1,3,3)
%e A351011   235:    11101011  (1,1,2,2,1,1)
%e A351011   528:  1000010000  (5,5)
%e A351011   547:  1000100011  (4,4,1,1)
%e A351011   586:  1001001010  (3,3,2,2)
%e A351011   676:  1010100100  (2,2,3,3)
%e A351011   698:  1010111010  (2,2,1,1,2,2)
%e A351011   904:  1110001000  (1,1,4,4)
%e A351011   915:  1110010011  (1,1,3,3,1,1)
%t A351011 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A351011 Select[Range[0,1000],And@@(#==2&)/@Length/@Split[stc[#]]&]
%Y A351011 The case of twins (binary weight 2) is A000120.
%Y A351011 All terms are evil numbers A001969.
%Y A351011 These compositions are counted by A003242 interspersed with 0's.
%Y A351011 Partitions of this type are counted by A035457, any length A351005.
%Y A351011 The Heinz numbers of these compositions are A062503.
%Y A351011 Taking singles instead of twins gives A333489, complement A348612.
%Y A351011 This is the anti-run case of A351010.
%Y A351011 The strict case (distinct twins) is A351009, counted by A077957(n-2).
%Y A351011 A011782 counts compositions.
%Y A351011 A085207/A085208 represent concatenation of standard compositions.
%Y A351011 A345167 ranks alternating compositions, counted by A025047.
%Y A351011 A350355 ranks up/down compositions, counted by A025048.
%Y A351011 A350356 ranks down/up compositions, counted by A025049.
%Y A351011 A351014 counts distinct runs in standard compositions.
%Y A351011 Cf. A008965, A018819, A027383, A032020, A035363, A088218, A106356, A122129, A122134, A238279, A351007.
%Y A351011 Selected statistics of standard compositions:
%Y A351011 - Length is A000120.
%Y A351011 - Sum is A070939.
%Y A351011 - Heinz number is A333219.
%Y A351011 - Number of distinct parts is A334028.
%Y A351011 Selected classes of standard compositions:
%Y A351011 - Partitions are A114994, strict A333256.
%Y A351011 - Multisets are A225620, strict A333255.
%Y A351011 - Strict compositions are A233564.
%Y A351011 - Constant compositions are A272919.
%K A351011 nonn
%O A351011 1,2
%A A351011 _Gus Wiseman_, Feb 03 2022