cp's OEIS Frontend

A349153 Numbers k such that the k-th composition in standard order has sum equal to twice its reverse-alternating sum.

%I A349153 #9 Nov 22 2021 08:08:36
%S A349153 0,11,12,14,133,138,143,148,155,158,160,168,179,182,188,195,198,204,
%T A349153 208,216,227,230,236,240,248,2057,2066,2071,2077,2084,2091,2094,2101,
%U A349153 2106,2111,2120,2131,2134,2140,2149,2154,2159,2164,2171,2174,2192,2211,2214
%N A349153 Numbers k such that the k-th composition in standard order has sum equal to twice its reverse-alternating sum.
%C A349153 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.
%C A349153 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%e A349153 The terms and corresponding compositions begin:
%e A349153     0: ()
%e A349153    11: (2,1,1)
%e A349153    12: (1,3)
%e A349153    14: (1,1,2)
%e A349153   133: (5,2,1)
%e A349153   138: (4,2,2)
%e A349153   143: (4,1,1,1,1)
%e A349153   148: (3,2,3)
%e A349153   155: (3,1,2,1,1)
%e A349153   158: (3,1,1,1,2)
%e A349153   160: (2,6)
%e A349153   168: (2,2,4)
%e A349153   179: (2,1,3,1,1)
%e A349153   182: (2,1,2,1,2)
%e A349153   188: (2,1,1,1,3)
%t A349153 stc[n_]:=Differences[ Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A349153 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A349153 Select[Range[0,1000],Total[stc[#]]==2*sats[stc[#]]&]
%Y A349153 These compositions are counted by A262977 up to 0's.
%Y A349153 Except for 0, a subset of A345917.
%Y A349153 The unreversed version is A348614.
%Y A349153 The unreversed negative version is A349154.
%Y A349153 The negative version is A349155.
%Y A349153 A non-reverse unordered version is A349159, counted by A000712 up to 0's.
%Y A349153 An unordered version is A349160, counted by A006330 up to 0's.
%Y A349153 A003242 counts Carlitz compositions.
%Y A349153 A011782 counts compositions.
%Y A349153 A025047 counts alternating or wiggly compositions, complement A345192.
%Y A349153 A034871, A097805, and A345197 count compositions by alternating sum.
%Y A349153 A103919 counts partitions by alternating sum, reverse A344612.
%Y A349153 A116406 counts compositions with alternating sum >=0, ranked by A345913.
%Y A349153 A138364 counts compositions with alternating sum 0, ranked by A344619.
%Y A349153 Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A294175.
%Y A349153 Statistics of standard compositions:
%Y A349153 - The compositions themselves are the rows of A066099.
%Y A349153 - Number of parts is given by A000120, distinct A334028.
%Y A349153 - Sum and product of parts are given by A070939 and A124758.
%Y A349153 - Maximum and minimum parts are given by A333766 and A333768.
%Y A349153 - Heinz number is given by A333219.
%Y A349153 Classes of standard compositions:
%Y A349153 - Partitions and strict partitions are ranked by A114994 and A333256.
%Y A349153 - Multisets and sets are ranked by A225620 and A333255.
%Y A349153 - Strict and constant compositions are ranked by A233564 and A272919.
%Y A349153 - Carlitz compositions are ranked by A333489, complement A348612.
%Y A349153 - Alternating compositions are ranked by A345167, complement A345168.
%K A349153 nonn
%O A349153 1,2
%A A349153 _Gus Wiseman_, Nov 17 2021