cp's OEIS Frontend

A345912 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum -1.

%I A345912 #7 Jul 10 2021 03:05:04
%S A345912 5,18,23,25,29,68,75,78,81,85,90,95,98,103,105,109,114,119,121,125,
%T A345912 264,275,278,284,289,293,298,303,308,315,318,322,327,329,333,338,343,
%U A345912 345,349,356,363,366,369,373,378,383,388,395,398,401,405,410,415,418,423
%N A345912 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum -1.
%C A345912 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C A345912 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 A345912 The sequence of terms together with the corresponding compositions begins:
%e A345912       5: (2,1)
%e A345912      18: (3,2)
%e A345912      23: (2,1,1,1)
%e A345912      25: (1,3,1)
%e A345912      29: (1,1,2,1)
%e A345912      68: (4,3)
%e A345912      75: (3,2,1,1)
%e A345912      78: (3,1,1,2)
%e A345912      81: (2,4,1)
%e A345912      85: (2,2,2,1)
%e A345912      90: (2,1,2,2)
%e A345912      95: (2,1,1,1,1,1)
%e A345912      98: (1,4,2)
%e A345912     103: (1,3,1,1,1)
%e A345912     105: (1,2,3,1)
%t A345912 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345912 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A345912 Select[Range[0,100],sats[stc[#]]==-1&]
%Y A345912 These compositions are counted by A001791.
%Y A345912 These are the positions of -1's in A344618.
%Y A345912 The non-reverse version is A345910.
%Y A345912 The opposite (positive 1) version is A345911.
%Y A345912 The version for Heinz numbers of partitions is A345959.
%Y A345912 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345912 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345912 A011782 counts compositions.
%Y A345912 A097805 counts compositions by alternating or reverse-alternating sum.
%Y A345912 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345912 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345912 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A345912 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A345912 A345197 counts compositions by sum, length, and alternating sum.
%Y A345912 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345912 - k = 0:  counted by A088218, ranked by A344619/A344619.
%Y A345912 - k = 1:  counted by A000984, ranked by A345909/A345911.
%Y A345912 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345912 - k = 2:  counted by A088218, ranked by A345925/A345922.
%Y A345912 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345912 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345912 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345912 - k > 0:  counted by A027306, ranked by A345917/A345918.
%Y A345912 - k < 0:  counted by A294175, ranked by A345919/A345920.
%Y A345912 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345912 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345912 - k odd:  counted by A000302, ranked by A053738/A053738.
%Y A345912 Cf. A000070, A000346, A001105, A008549, A025047, A031444, A034871, A114121, A126869, A344608, A345958, A345959.
%K A345912 nonn
%O A345912 1,1
%A A345912 _Gus Wiseman_, Jul 01 2021