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 A345918 #7 Jul 10 2021 07:59:38
%S A345918 1,2,4,6,7,8,11,12,14,16,19,20,21,22,24,26,27,28,30,31,32,35,37,38,40,
%T A345918 42,44,47,48,51,52,54,56,59,60,62,64,67,69,70,72,73,74,76,79,80,82,83,
%U A345918 84,86,87,88,91,92,93,94,96,99,100,101,102,104,106,107,108
%N A345918 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum > 0.
%C A345918 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C A345918 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 A345918 The initial terms and the corresponding compositions:
%e A345918 1: (1) 26: (1,2,2) 52: (1,2,3)
%e A345918 2: (2) 27: (1,2,1,1) 54: (1,2,1,2)
%e A345918 4: (3) 28: (1,1,3) 56: (1,1,4)
%e A345918 6: (1,2) 30: (1,1,1,2) 59: (1,1,2,1,1)
%e A345918 7: (1,1,1) 31: (1,1,1,1,1) 60: (1,1,1,3)
%e A345918 8: (4) 32: (6) 62: (1,1,1,1,2)
%e A345918 11: (2,1,1) 35: (4,1,1) 64: (7)
%e A345918 12: (1,3) 37: (3,2,1) 67: (5,1,1)
%e A345918 14: (1,1,2) 38: (3,1,2) 69: (4,2,1)
%e A345918 16: (5) 40: (2,4) 70: (4,1,2)
%e A345918 19: (3,1,1) 42: (2,2,2) 72: (3,4)
%e A345918 20: (2,3) 44: (2,1,3) 73: (3,3,1)
%e A345918 21: (2,2,1) 47: (2,1,1,1,1) 74: (3,2,2)
%e A345918 22: (2,1,2) 48: (1,5) 76: (3,1,3)
%e A345918 24: (1,4) 51: (1,3,1,1) 79: (3,1,1,1,1)
%t A345918 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345918 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A345918 Select[Range[0,100],sats[stc[#]]>0&]
%Y A345918 The version for prime indices is A000037.
%Y A345918 The version for Heinz numbers of partitions is A026424, counted by A027193.
%Y A345918 These compositions are counted by A027306.
%Y A345918 These are the positions of terms > 0 in A344618.
%Y A345918 The weak (k >= 0) version is A345914.
%Y A345918 The version for unreversed alternating sum is A345917.
%Y A345918 The opposite (k < 0) version is A345920.
%Y A345918 A011782 counts compositions.
%Y A345918 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345918 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345918 A236913 counts partitions of 2n with reverse-alternating sum <= 0.
%Y A345918 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345918 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A345918 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A345918 A345197 counts compositions by sum, length, and alternating sum.
%Y A345918 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345918 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345918 - k = 0: counted by A088218, ranked by A344619/A344619.
%Y A345918 - k = 1: counted by A000984, ranked by A345909/A345911.
%Y A345918 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345918 - k = 2: counted by A088218, ranked by A345925/A345922.
%Y A345918 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345918 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345918 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345918 - k > 0: counted by A027306, ranked by A345917/A345918.
%Y A345918 - k < 0: counted by A294175, ranked by A345919/A345920.
%Y A345918 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345918 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345918 - k odd: counted by A000302, ranked by A053738/A053738.
%Y A345918 Cf. A000070, A000290, A000346, A008549, A025047, A027187, A028260, A034871, A114121, A163493, A344607, A344608, A344650, A344743, A345908.
%K A345918 nonn
%O A345918 1,2
%A A345918 _Gus Wiseman_, Jul 09 2021