cp's OEIS Frontend

A348614 Numbers k such that the k-th composition in standard order has sum equal to twice its alternating sum.

%I A348614 #8 Oct 31 2021 01:40:40
%S A348614 0,9,11,14,130,133,135,138,141,143,148,153,155,158,168,177,179,182,
%T A348614 188,208,225,227,230,236,248,2052,2057,2059,2062,2066,2069,2071,2074,
%U A348614 2077,2079,2084,2089,2091,2094,2098,2101,2103,2106,2109,2111,2120,2129,2131
%N A348614 Numbers k such that the k-th composition in standard order has sum equal to twice its alternating sum.
%C A348614 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 A348614 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%H A348614 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e A348614 The terms together with their binary indices begin:
%e A348614     0: ()
%e A348614     9: (3,1)
%e A348614    11: (2,1,1)
%e A348614    14: (1,1,2)
%e A348614   130: (6,2)
%e A348614   133: (5,2,1)
%e A348614   135: (5,1,1,1)
%e A348614   138: (4,2,2)
%e A348614   141: (4,1,2,1)
%e A348614   143: (4,1,1,1,1)
%e A348614   148: (3,2,3)
%e A348614   153: (3,1,3,1)
%e A348614   155: (3,1,2,1,1)
%e A348614   158: (3,1,1,1,2)
%t A348614 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A348614 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A348614 Select[Range[0,1000],Total[stc[#]]==2*ats[stc[#]]&]
%Y A348614 The unordered case (partitions) is counted by A000712, reverse A006330.
%Y A348614 These compositions are counted by A262977.
%Y A348614 Except for 0, a subset of A345917 (which is itself a subset of A345913).
%Y A348614 A000346 = even-length compositions with alt sum != 0, complement A001700.
%Y A348614 A011782 counts compositions.
%Y A348614 A025047 counts wiggly compositions, ranked by A345167.
%Y A348614 A034871 counts compositions of 2n with alternating sum 2k.
%Y A348614 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A348614 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A348614 A116406 counts compositions with alternating sum >=0, ranked by A345913.
%Y A348614 A138364 counts compositions with alternating sum 0, ranked by A344619.
%Y A348614 A345197 counts compositions by length and alternating sum.
%Y A348614 Cf. A000984, A002458, A004331, A005810, A224274.
%Y A348614 Cf. A008549, A013777, A027306, A058622, A088218, A114121, A120452, A126869, A163493, A294175, A344604.
%K A348614 nonn
%O A348614 1,2
%A A348614 _Gus Wiseman_, Oct 29 2021