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 A345913 #7 Jul 10 2021 03:05:16
%S A345913 0,1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,18,19,21,22,23,26,28,29,31,32,
%T A345913 33,34,35,36,37,38,39,41,42,43,44,45,46,47,50,52,53,55,56,57,58,59,61,
%U A345913 62,63,64,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82
%N A345913 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum >= 0.
%C A345913 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C A345913 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 A345913 The sequence of terms together with the corresponding compositions begins:
%e A345913 0: () 17: (4,1) 37: (3,2,1)
%e A345913 1: (1) 18: (3,2) 38: (3,1,2)
%e A345913 2: (2) 19: (3,1,1) 39: (3,1,1,1)
%e A345913 3: (1,1) 21: (2,2,1) 41: (2,3,1)
%e A345913 4: (3) 22: (2,1,2) 42: (2,2,2)
%e A345913 5: (2,1) 23: (2,1,1,1) 43: (2,2,1,1)
%e A345913 7: (1,1,1) 26: (1,2,2) 44: (2,1,3)
%e A345913 8: (4) 28: (1,1,3) 45: (2,1,2,1)
%e A345913 9: (3,1) 29: (1,1,2,1) 46: (2,1,1,2)
%e A345913 10: (2,2) 31: (1,1,1,1,1) 47: (2,1,1,1,1)
%e A345913 11: (2,1,1) 32: (6) 50: (1,3,2)
%e A345913 13: (1,2,1) 33: (5,1) 52: (1,2,3)
%e A345913 14: (1,1,2) 34: (4,2) 53: (1,2,2,1)
%e A345913 15: (1,1,1,1) 35: (4,1,1) 55: (1,2,1,1,1)
%e A345913 16: (5) 36: (3,3) 56: (1,1,4)
%t A345913 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345913 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345913 Select[Range[0,100],ats[stc[#]]>=0&]
%Y A345913 These compositions are counted by A116406.
%Y A345913 These are the positions of terms >= 0 in A124754.
%Y A345913 The version for prime indices is A344609.
%Y A345913 The reverse-alternating sum version is A345914.
%Y A345913 The opposite (k <= 0) version is A345915.
%Y A345913 The strict (k > 0) version is A345917.
%Y A345913 The complement is A345919.
%Y A345913 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345913 A011782 counts compositions.
%Y A345913 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345913 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345913 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345913 A345197 counts compositions by sum, length, and alternating sum.
%Y A345913 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345913 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345913 - k = 0: counted by A088218, ranked by A344619/A344619.
%Y A345913 - k = 1: counted by A000984, ranked by A345909/A345911.
%Y A345913 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345913 - k = 2: counted by A088218, ranked by A345925/A345922.
%Y A345913 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345913 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345913 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345913 - k > 0: counted by A027306, ranked by A345917/A345918.
%Y A345913 - k < 0: counted by A294175, ranked by A345919/A345920.
%Y A345913 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345913 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345913 - k odd: counted by A000302, ranked by A053738/A053738.
%Y A345913 Cf. A000070, A000346, A008549, A025047, A027187, A032443, A034871, A114121, A163493, A236913, A238279, A344607, A344608, A344610, A344611.
%K A345913 nonn
%O A345913 1,3
%A A345913 _Gus Wiseman_, Jul 04 2021