OFFSET
1,2
COMMENTS
We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...
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
John Tyler Rascoe, Table of n, a(n) for n = 1..7762
EXAMPLE
The sequence together with the corresponding compositions begins:
0: ()
14: (1,1,2)
15: (1,1,1,1)
44: (2,1,3)
45: (2,1,2,1)
46: (2,1,1,2)
52: (1,2,3)
53: (1,2,2,1)
54: (1,2,1,2)
59: (1,1,2,1,1)
61: (1,1,1,2,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Select[Range[0, 100], halfats[stc[#]]==0&]
PROG
(Python)
from itertools import count, islice
def comp(n): #row n of A066099 after Franklin T. Adams-Watters
v, k = [], 0
while n > 0:
k += 1
if n%2 == 1:
v.append(k)
k = 0
n = n//2
return(v[::-1])
def a_gen():
for n in count(0):
c = comp(n)
x = sum(c[i]*(-1)**(i//2) for i in range(len(c)))
if x == 0:
yield(n)
A357625_list = list(islice(a_gen(), 60)) # John Tyler Rascoe, Jun 01 2024
CROSSREFS
See link for sequences related to standard compositions.
The version for full alternating sum is A344619.
Positions of zeros in A357621.
The reverse version is A357626.
The version for prime indices is A357631.
The version for Heinz numbers of partitions is A357635.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 08 2022
STATUS
approved