A345913 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum >= 0.

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, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82

Offset: 1

Author

Gus Wiseman, Jul 04 2021

Keywords

• nonn

Comments

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) 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 sequence of terms together with the corresponding compositions begins:
     0: ()           17: (4,1)          37: (3,2,1)
     1: (1)          18: (3,2)          38: (3,1,2)
     2: (2)          19: (3,1,1)        39: (3,1,1,1)
     3: (1,1)        21: (2,2,1)        41: (2,3,1)
     4: (3)          22: (2,1,2)        42: (2,2,2)
     5: (2,1)        23: (2,1,1,1)      43: (2,2,1,1)
     7: (1,1,1)      26: (1,2,2)        44: (2,1,3)
     8: (4)          28: (1,1,3)        45: (2,1,2,1)
     9: (3,1)        29: (1,1,2,1)      46: (2,1,1,2)
    10: (2,2)        31: (1,1,1,1,1)    47: (2,1,1,1,1)
    11: (2,1,1)      32: (6)            50: (1,3,2)
    13: (1,2,1)      33: (5,1)          52: (1,2,3)
    14: (1,1,2)      34: (4,2)          53: (1,2,2,1)
    15: (1,1,1,1)    35: (4,1,1)        55: (1,2,1,1,1)
    16: (5)          36: (3,3)          56: (1,1,4)
		

Crossrefs

These compositions are counted by A116406.

These are the positions of terms >= 0 in A124754.

The version for prime indices is A344609.

The reverse-alternating sum version is A345914.

The opposite (k <= 0) version is A345915.

The strict (k > 0) version is A345917.

The complement is A345919.

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).

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

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, A000346, A008549, A025047, A027187, A032443, A034871, A114121, A163493, A236913, A238279, A344607, A344608, A344610, A344611.

Programs

Mathematica

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