cp's OEIS Frontend

A349155 Numbers k such that the k-th composition in standard order has sum equal to negative twice its reverse-alternating sum.

%I A349155 #7 Nov 22 2021 19:59:46
%S A349155 0,9,130,135,141,153,177,193,225,2052,2059,2062,2069,2074,2079,2089,
%T A349155 2098,2103,2109,2129,2146,2151,2157,2169,2209,2242,2247,2253,2265,
%U A349155 2289,2369,2434,2439,2445,2457,2481,2529,2561,2689,2818,2823,2829,2841,2865,2913
%N A349155 Numbers k such that the k-th composition in standard order has sum equal to negative twice its reverse-alternating sum.
%C A349155 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 A349155 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%e A349155 The terms and corresponding compositions begin:
%e A349155      0: ()
%e A349155      9: (3,1)
%e A349155    130: (6,2)
%e A349155    135: (5,1,1,1)
%e A349155    141: (4,1,2,1)
%e A349155    153: (3,1,3,1)
%e A349155    177: (2,1,4,1)
%e A349155    193: (1,6,1)
%e A349155    225: (1,1,5,1)
%e A349155   2052: (9,3)
%e A349155   2059: (8,2,1,1)
%e A349155   2062: (8,1,1,2)
%e A349155   2069: (7,2,2,1)
%e A349155   2074: (7,1,2,2)
%e A349155   2079: (7,1,1,1,1,1)
%e A349155   2089: (6,2,3,1)
%e A349155   2098: (6,1,3,2)
%e A349155   2103: (6,1,2,1,1,1)
%t A349155 stc[n_]:=Differences[Prepend[ Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A349155 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A349155 Select[Range[0,1000],Total[stc[#]]==-2*sats[stc[#]]&]
%Y A349155 These compositions are counted by A224274 up to 0's.
%Y A349155 An unordered version is A348617, counted by A001523 up to 0's.
%Y A349155 The positive version is A349153, unreversed A348614.
%Y A349155 The unreversed version is A349154.
%Y A349155 Positive unordered unreversed: A349159, counted by A000712 up to 0's.
%Y A349155 A positive unordered version is A349160, counted by A006330 up to 0's.
%Y A349155 A003242 counts Carlitz compositions.
%Y A349155 A011782 counts compositions.
%Y A349155 A025047 counts alternating or wiggly compositions, complement A345192.
%Y A349155 A034871, A097805, and A345197 count compositions by alternating sum.
%Y A349155 A103919 counts partitions by alternating sum, reverse A344612.
%Y A349155 A116406 counts compositions with alternating sum >=0, ranked by A345913.
%Y A349155 A138364 counts compositions with alternating sum 0, ranked by A344619.
%Y A349155 Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917.
%Y A349155 Statistics of standard compositions:
%Y A349155 - The compositions themselves are the rows of A066099.
%Y A349155 - Number of parts is given by A000120, distinct A334028.
%Y A349155 - Sum and product of parts are given by A070939 and A124758.
%Y A349155 - Maximum and minimum parts are given by A333766 and A333768.
%Y A349155 - Heinz number is given by A333219.
%Y A349155 Classes of standard compositions:
%Y A349155 - Partitions and strict partitions are ranked by A114994 and A333256.
%Y A349155 - Multisets and sets are ranked by A225620 and A333255.
%Y A349155 - Strict and constant compositions are ranked by A233564 and A272919.
%Y A349155 - Carlitz compositions are ranked by A333489, complement A348612.
%Y A349155 - Alternating compositions are ranked by A345167, complement A345168.
%K A349155 nonn
%O A349155 1,2
%A A349155 _Gus Wiseman_, Nov 22 2021