|
|
A345909
|
|
Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 1.
|
|
29
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
|
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 version for Heinz numbers of partitions is A345958.
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:
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|