

A357184


Numbers k such that the kth composition in standard order has the same length as its alternating sum.


20



0, 1, 9, 19, 22, 28, 34, 69, 74, 84, 104, 132, 135, 141, 153, 177, 225, 265, 271, 274, 283, 286, 292, 307, 310, 316, 328, 355, 358, 364, 376, 400, 451, 454, 460, 472, 496, 520, 523, 526, 533, 538, 553, 562, 593, 610, 673, 706, 833, 898, 1041, 1047, 1053, 1058
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OFFSET

1,3


COMMENTS

A composition of n is a finite sequence of positive integers summing to n. The kth composition in standard order (graded reverselexicographic, 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.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (1)^(i1) y_i.


LINKS



EXAMPLE

The sequence together with the corresponding compositions begins:
0: ()
1: (1)
9: (3,1)
19: (3,1,1)
22: (2,1,2)
28: (1,1,3)
34: (4,2)
69: (4,2,1)
74: (3,2,2)
84: (2,2,3)
104: (1,2,4)
132: (5,3)
135: (5,1,1,1)
141: (4,1,2,1)
153: (3,1,3,1)
177: (2,1,4,1)
225: (1,1,5,1)


MATHEMATICA

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
ats[y_]:=Sum[(1)^(i1)*y[[i]], {i, Length[y]}];
Select[Range[0, 100], Length[stc[#]]==ats[stc[#]]&]


CROSSREFS

See link for sequences related to standard compositions.
For sum equal to twice alternating sum we have A348614, counted by A262977.
These compositions are counted by A357182.
The case of partitions is counted by A357189.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A357136 counts compositions by alternating sum.


KEYWORD

nonn


AUTHOR



STATUS

approved



