This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A345920 #6 Jul 10 2021 07:59:53
%S A345920 5,9,17,18,23,25,29,33,34,39,45,49,57,65,66,68,71,75,77,78,81,85,89,
%T A345920 90,95,97,98,103,105,109,113,114,119,121,125,129,130,132,135,139,141,
%U A345920 142,149,153,154,159,161,169,177,178,183,189,193,194,199,205,209,217
%N A345920 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum < 0.
%C A345920 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C A345920 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 A345920 The initial terms and the corresponding compositions:
%e A345920 5: (2,1) 68: (4,3)
%e A345920 9: (3,1) 71: (4,1,1,1)
%e A345920 17: (4,1) 75: (3,2,1,1)
%e A345920 18: (3,2) 77: (3,1,2,1)
%e A345920 23: (2,1,1,1) 78: (3,1,1,2)
%e A345920 25: (1,3,1) 81: (2,4,1)
%e A345920 29: (1,1,2,1) 85: (2,2,2,1)
%e A345920 33: (5,1) 89: (2,1,3,1)
%e A345920 34: (4,2) 90: (2,1,2,2)
%e A345920 39: (3,1,1,1) 95: (2,1,1,1,1,1)
%e A345920 45: (2,1,2,1) 97: (1,5,1)
%e A345920 49: (1,4,1) 98: (1,4,2)
%e A345920 57: (1,1,3,1) 103: (1,3,1,1,1)
%e A345920 65: (6,1) 105: (1,2,3,1)
%e A345920 66: (5,2) 109: (1,2,1,2,1)
%t A345920 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345920 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A345920 Select[Range[0,100],sats[stc[#]]<0&]
%Y A345920 The version for prime indices is {}.
%Y A345920 The version for Heinz numbers of partitions is A119899.
%Y A345920 These compositions are counted by A294175 (even bisection: A008549).
%Y A345920 These are the positions of terms < 0 in A344618.
%Y A345920 The complement is A345914.
%Y A345920 The weak (k <= 0) version is A345916.
%Y A345920 The opposite (k > 0) version is A345918.
%Y A345920 The version for unreversed alternating sum is A345919.
%Y A345920 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345920 A011782 counts compositions.
%Y A345920 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345920 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345920 A236913 counts partitions of 2n with reverse-alternating sum <= 0.
%Y A345920 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345920 A345197 counts compositions by sum, length, and alternating sum.
%Y A345920 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345920 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345920 - k = 0: counted by A088218, ranked by A344619/A344619.
%Y A345920 - k = 1: counted by A000984, ranked by A345909/A345911.
%Y A345920 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345920 - k = 2: counted by A088218, ranked by A345925/A345922.
%Y A345920 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345920 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345920 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345920 - k > 0: counted by A027306, ranked by A345917/A345918.
%Y A345920 - k < 0: counted by A294175, ranked by A345919/A345920.
%Y A345920 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345920 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345920 - k odd: counted by A000302, ranked by A053738/A053738.
%Y A345920 Cf. A000070, A000346, A025047, A028260, A032443, A034871, A106356, A114121, A163493, A344608, A344610, A344611, A345908.
%K A345920 nonn
%O A345920 1,1
%A A345920 _Gus Wiseman_, Jul 09 2021