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).

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.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8118 (rows for n = 0..1023 flattened)

Index entries for sequences related to binary expansion of n

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.

Cf. A003714, A037800, A277561, A374355, A374356.

Sequence in context: A379016 A276669 A307596 * A373043 A240205 A340732

Adjacent sequences: A374351 A374352 A374353 * A374355 A374356 A374357

Keyword

nonn,base,tabf

Author

Rémy Sigrist, Jul 06 2024

Status

approved