A345923 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum -2.

9, 34, 39, 45, 49, 57, 132, 139, 142, 149, 154, 159, 161, 169, 178, 183, 189, 194, 199, 205, 209, 217, 226, 231, 237, 241, 249, 520, 531, 534, 540, 549, 554, 559, 564, 571, 574, 577, 585, 594, 599, 605, 612, 619, 622, 629, 634, 639, 642, 647, 653, 657, 665

Offset: 1

Author

Gus Wiseman, Jul 10 2021

Keywords

• nonn

Comments

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

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.

Examples

			The initial terms and the corresponding compositions:
      9: (3,1)            183: (2,1,2,1,1,1)
     34: (4,2)            189: (2,1,1,1,2,1)
     39: (3,1,1,1)        194: (1,5,2)
     45: (2,1,2,1)        199: (1,4,1,1,1)
     49: (1,4,1)          205: (1,3,1,2,1)
     57: (1,1,3,1)        209: (1,2,4,1)
    132: (5,3)            217: (1,2,1,3,1)
    139: (4,2,1,1)        226: (1,1,4,2)
    142: (4,1,1,2)        231: (1,1,3,1,1,1)
    149: (3,2,2,1)        237: (1,1,2,1,2,1)
    154: (3,1,2,2)        241: (1,1,1,4,1)
    159: (3,1,1,1,1,1)    249: (1,1,1,1,3,1)
    161: (2,5,1)          520: (6,4)
    169: (2,2,3,1)        531: (5,3,1,1)
    178: (2,1,3,2)        534: (5,2,1,2)
		

Crossrefs

These compositions are counted by A088218.

These are the positions of 2's in A344618.

The case of partitions of 2n is A344741.

The opposite (negative 2) version is A345923.

The version for unreversed alternating sum is A345925.

The version for Heinz numbers of partitions is A345961.

A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

A011782 counts compositions.

A097805 counts compositions by alternating (or reverse-alternating) sum.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A120452 counts partitions of 2n with reverse-alternating sum 2.

A316524 gives the alternating sum of prime indices (reverse: A344616).

A344610 counts partitions by sum and positive reverse-alternating sum.

A344611 counts partitions of 2n with reverse-alternating sum >= 0.

A345197 counts compositions by sum, length, and alternating sum.

Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.

Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:

- k = 0: counted by A088218, ranked by A344619/A344619.

- k = 1: counted by A000984, ranked by A345909/A345911.

- k = -1: counted by A001791, ranked by A345910/A345912.

- k = 2: counted by A088218, ranked by A345925/A345922.

- k = -2: counted by A002054, ranked by A345924/A345923.

- k >= 0: counted by A116406, ranked by A345913/A345914.

- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.

- k > 0: counted by A027306, ranked by A345917/A345918.

- k < 0: counted by A294175, ranked by A345919/A345920.

- k != 0: counted by A058622, ranked by A345921/A345921.

- k even: counted by A081294, ranked by A053754/A053754.

- k odd: counted by A000302, ranked by A053738/A053738.

Cf. A000070, A000097, A000346, A025047, A032443, A034871, A106356, A163493, A236913, A344607, A344608, A344743.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]==-2&]