cp's OEIS Frontend

A345909 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 1.

%I A345909 #9 Jul 08 2021 14:23:17
%S A345909 1,5,7,18,21,23,26,29,31,68,73,75,78,82,85,87,90,93,95,100,105,107,
%T A345909 110,114,117,119,122,125,127,264,273,275,278,284,290,293,295,298,301,
%U A345909 303,308,313,315,318,324,329,331,334,338,341,343,346,349,351,356,361
%N A345909 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 1.
%C A345909 The alternating sum of a composition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C A345909 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 A345909 The sequence of terms together with the corresponding compositions begins:
%e A345909       1: (1)             87: (2,2,1,1,1)
%e A345909       5: (2,1)           90: (2,1,2,2)
%e A345909       7: (1,1,1)         93: (2,1,1,2,1)
%e A345909      18: (3,2)           95: (2,1,1,1,1,1)
%e A345909      21: (2,2,1)        100: (1,3,3)
%e A345909      23: (2,1,1,1)      105: (1,2,3,1)
%e A345909      26: (1,2,2)        107: (1,2,2,1,1)
%e A345909      29: (1,1,2,1)      110: (1,2,1,1,2)
%e A345909      31: (1,1,1,1,1)    114: (1,1,3,2)
%e A345909      68: (4,3)          117: (1,1,2,2,1)
%e A345909      73: (3,3,1)        119: (1,1,2,1,1,1)
%e A345909      75: (3,2,1,1)      122: (1,1,1,2,2)
%e A345909      78: (3,1,1,2)      125: (1,1,1,1,2,1)
%e A345909      82: (2,3,2)        127: (1,1,1,1,1,1,1)
%e A345909      85: (2,2,2,1)      264: (5,4)
%t A345909 stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345909 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345909 Select[Range[0,100],ats[stc[#]]==1&]
%Y A345909 These compositions are counted by A000984 (bisection of A126869).
%Y A345909 The version for prime indices is A001105.
%Y A345909 A version using runs of binary digits is A031448.
%Y A345909 These are the positions of 1's in A124754.
%Y A345909 The opposite (negative 1) version is A345910.
%Y A345909 The reverse version is A345911.
%Y A345909 The version for Heinz numbers of partitions is A345958.
%Y A345909 Standard compositions: A000120, A066099, A070939, A124754, A228351, A344618.
%Y A345909 A000070 counts partitions with alternating sum 1 (ranked by A345957).
%Y A345909 A000097 counts partitions with alternating sum 2 (ranked by A345960).
%Y A345909 A011782 counts compositions.
%Y A345909 A097805 counts compositions by sum and alternating sum.
%Y A345909 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345909 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345909 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A345909 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A345909 A345197 counts compositions by sum, length, and alternating sum.
%Y A345909 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345909 - k = 0:  counted by A088218, ranked by A344619/A344619.
%Y A345909 - k = 1:  counted by A000984, ranked by A345909 (this sequence)/A345911.
%Y A345909 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345909 - k = 2:  counted by A088218, ranked by A345925/A345922.
%Y A345909 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345909 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345909 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345909 - k > 0:  counted by A027306, ranked by A345917/A345918.
%Y A345909 - k < 0:  counted by A294175, ranked by A345919/A345920.
%Y A345909 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345909 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345909 - k odd:  counted by A000302, ranked by A053738/A053738.
%Y A345909 Cf. A000346, A008549, A025047, A031443, A031444, A034871, A114121, A238279, A304620, A344651.
%K A345909 nonn
%O A345909 1,2
%A A345909 _Gus Wiseman_, Jun 30 2021