

A349155


Numbers k such that the kth composition in standard order has sum equal to negative twice its reversealternating sum.


5



0, 9, 130, 135, 141, 153, 177, 193, 225, 2052, 2059, 2062, 2069, 2074, 2079, 2089, 2098, 2103, 2109, 2129, 2146, 2151, 2157, 2169, 2209, 2242, 2247, 2253, 2265, 2289, 2369, 2434, 2439, 2445, 2457, 2481, 2529, 2561, 2689, 2818, 2823, 2829, 2841, 2865, 2913
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OFFSET

1,2


COMMENTS

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 reversealternating sum of a sequence (y_1,...,y_k) is Sum_i (1)^(ki) y_i.


LINKS



EXAMPLE

The terms and corresponding compositions begin:
0: ()
9: (3,1)
130: (6,2)
135: (5,1,1,1)
141: (4,1,2,1)
153: (3,1,3,1)
177: (2,1,4,1)
193: (1,6,1)
225: (1,1,5,1)
2052: (9,3)
2059: (8,2,1,1)
2062: (8,1,1,2)
2069: (7,2,2,1)
2074: (7,1,2,2)
2079: (7,1,1,1,1,1)
2089: (6,2,3,1)
2098: (6,1,3,2)
2103: (6,1,2,1,1,1)


MATHEMATICA

stc[n_]:=Differences[Prepend[ Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
sats[y_]:=Sum[(1)^(iLength[y])*y[[i]], {i, Length[y]}];
Select[Range[0, 1000], Total[stc[#]]==2*sats[stc[#]]&]


CROSSREFS

These compositions are counted by A224274 up to 0's.
A positive unordered version is A349160, counted by A006330 up to 0's.
A003242 counts Carlitz compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917.
Statistics of standard compositions:
 The compositions themselves are the rows of A066099.
 Heinz number is given by A333219.
Classes of standard compositions:


KEYWORD

nonn


AUTHOR



STATUS

approved



