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 A345921 #6 Jul 12 2021 18:01:52
%S A345921 1,2,4,5,6,7,8,9,11,12,14,16,17,18,19,20,21,22,23,24,25,26,27,28,29,
%T A345921 30,31,32,33,34,35,37,38,39,40,42,44,45,47,48,49,51,52,54,56,57,59,60,
%U A345921 62,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81
%N A345921 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum != 0.
%C A345921 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C A345921 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 A345921 Also numbers k such that the k-th composition in standard order has reverse-alternating sum != 0.
%e A345921 The initial terms and the corresponding compositions:
%e A345921 1: (1) 20: (2,3) 35: (4,1,1)
%e A345921 2: (2) 21: (2,2,1) 37: (3,2,1)
%e A345921 4: (3) 22: (2,1,2) 38: (3,1,2)
%e A345921 5: (2,1) 23: (2,1,1,1) 39: (3,1,1,1)
%e A345921 6: (1,2) 24: (1,4) 40: (2,4)
%e A345921 7: (1,1,1) 25: (1,3,1) 42: (2,2,2)
%e A345921 8: (4) 26: (1,2,2) 44: (2,1,3)
%e A345921 9: (3,1) 27: (1,2,1,1) 45: (2,1,2,1)
%e A345921 11: (2,1,1) 28: (1,1,3) 47: (2,1,1,1,1)
%e A345921 12: (1,3) 29: (1,1,2,1) 48: (1,5)
%e A345921 14: (1,1,2) 30: (1,1,1,2) 49: (1,4,1)
%e A345921 16: (5) 31: (1,1,1,1,1) 51: (1,3,1,1)
%e A345921 17: (4,1) 32: (6) 52: (1,2,3)
%e A345921 18: (3,2) 33: (5,1) 54: (1,2,1,2)
%e A345921 19: (3,1,1) 34: (4,2) 56: (1,1,4)
%t A345921 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345921 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345921 Select[Range[0,100],ats[stc[#]]!=0&]
%Y A345921 The version for Heinz numbers of partitions is A000037.
%Y A345921 These compositions are counted by A058622.
%Y A345921 These are the positions of terms != 0 in A124754.
%Y A345921 The complement (k = 0) is A344619.
%Y A345921 The positive (k > 0) version is A345917 (reverse: A345918).
%Y A345921 The negative (k < 0) version is A345919 (reverse: A345920).
%Y A345921 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345921 A011782 counts compositions.
%Y A345921 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345921 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345921 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345921 A345197 counts compositions by sum, length, and alternating sum.
%Y A345921 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345921 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345921 - k = 0: counted by A088218, ranked by A344619/A344619.
%Y A345921 - k = 1: counted by A000984, ranked by A345909/A345911.
%Y A345921 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345921 - k = 2: counted by A088218, ranked by A345925/A345922.
%Y A345921 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345921 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345921 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345921 - k > 0: counted by A027306, ranked by A345917/A345918.
%Y A345921 - k < 0: counted by A294175, ranked by A345919/A345920.
%Y A345921 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345921 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345921 - k odd: counted by A000302, ranked by A053738/A053738.
%Y A345921 Cf. A000070, A000346, A008549, A025047, A032443, A034871, A114121, A163493, A236913, A344609, A344651, A345908.
%K A345921 nonn
%O A345921 1,2
%A A345921 _Gus Wiseman_, Jul 10 2021