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

1, 5, 7, 18, 21, 23, 26, 29, 31, 68, 73, 75, 78, 82, 85, 87, 90, 93, 95, 100, 105, 107, 110, 114, 117, 119, 122, 125, 127, 264, 273, 275, 278, 284, 290, 293, 295, 298, 301, 303, 308, 313, 315, 318, 324, 329, 331, 334, 338, 341, 343, 346, 349, 351, 356, 361

Offset

1,2

Comments

The alternating sum of a composition (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.

Links

Table of n, a(n) for n=1..56.

Example

The sequence of terms together with the corresponding compositions begins:
      1: (1)             87: (2,2,1,1,1)
      5: (2,1)           90: (2,1,2,2)
      7: (1,1,1)         93: (2,1,1,2,1)
     18: (3,2)           95: (2,1,1,1,1,1)
     21: (2,2,1)        100: (1,3,3)
     23: (2,1,1,1)      105: (1,2,3,1)
     26: (1,2,2)        107: (1,2,2,1,1)
     29: (1,1,2,1)      110: (1,2,1,1,2)
     31: (1,1,1,1,1)    114: (1,1,3,2)
     68: (4,3)          117: (1,1,2,2,1)
     73: (3,3,1)        119: (1,1,2,1,1,1)
     75: (3,2,1,1)      122: (1,1,1,2,2)
     78: (3,1,1,2)      125: (1,1,1,1,2,1)
     82: (2,3,2)        127: (1,1,1,1,1,1,1)
     85: (2,2,2,1)      264: (5,4)

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[#]]==1&]

Crossrefs

These compositions are counted by A000984 (bisection of A126869).

The version for prime indices is A001105.

A version using runs of binary digits is A031448.

These are the positions of 1's in A124754.

The opposite (negative 1) version is A345910.

The reverse version is A345911.

The version for Heinz numbers of partitions is A345958.

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

A000070 counts partitions with alternating sum 1 (ranked by A345957).

A000097 counts partitions with alternating sum 2 (ranked by A345960).

A011782 counts compositions.

A097805 counts compositions by sum and alternating sum.

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

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.

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 (this sequence)/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. A000346, A008549, A025047, A031443, A031444, A034871, A114121, A238279, A304620, A344651.

Sequence in context: A145582 A034762 A138919 * A214414 A153192 A263878

Adjacent sequences: A345906 A345907 A345908 * A345910 A345911 A345912

Keyword

nonn

Author

Gus Wiseman, Jun 30 2021

Status

approved