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A374354
Irregular table T(n, k), n >= 0, 0 <= k < A277561(n), read by rows; the n-th row lists the fibbinary numbers f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).
5
0, 0, 1, 0, 2, 1, 2, 0, 4, 0, 1, 4, 5, 2, 4, 2, 5, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 1, 2, 9, 10, 4, 8, 4, 5, 8, 9, 4, 10, 5, 10, 0, 16, 0, 1, 16, 17, 0, 2, 16, 18, 1, 2, 17, 18, 0, 4, 16, 20, 0, 1, 4, 5, 16, 17, 20, 21, 2, 4, 18, 20, 2, 5, 18, 21, 8, 16, 8, 9, 16, 17
OFFSET
0,5
COMMENTS
In other words, we partition n into pairs of fibbinary numbers whose binary expansions have no common 1's and list the corresponding fibbinary numbers to get the n-th row.
FORMULA
T(n, 0) = 0 iff n is a fibbinary number.
T(n, k) + T(n, A277561(n)-1-k) = n.
T(n, 0) = A374355(n).
T(n, A277561(n)-1) = A374356(n).
Sum_{k = 0..A277561(n)-1} T(n, k) = n * 2^A037800(n).
EXAMPLE
Triangle T(n, k) begins:
n n-th row
-- -----------
0 0
1 0, 1
2 0, 2
3 1, 2
4 0, 4
5 0, 1, 4, 5
6 2, 4
7 2, 5
8 0, 8
9 0, 1, 8, 9
10 0, 2, 8, 10
11 1, 2, 9, 10
12 4, 8
13 4, 5, 8, 9
14 4, 10
15 5, 10
16 0, 16
PROG
(PARI) row(n) = { my (r = [0], e, x, y, b); while (n, x = y = 0; e = valuation(n, 2); for (k = 0, oo, if (bittest(n, e+k), n -= b = 2^(e+k); [x, y] = [y + b, x], r = concat([v + y | v <- r], [v + x | v <- r]); break; ); ); ); return (r); }
CROSSREFS
See A295989 and A374361 for similar sequences.
Sequence in context: A138498 A276669 A307596 * A373043 A240205 A340732
KEYWORD
nonn,base,tabf
AUTHOR
Rémy Sigrist, Jul 06 2024
STATUS
approved