A301983 Irregular triangle read by rows T(n, k), n >= 1 and 1 <= k <= A301977(n): T(n, k) is the k-th positive number whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

1, 1, 2, 1, 3, 1, 2, 4, 1, 2, 3, 5, 1, 2, 3, 6, 1, 3, 7, 1, 2, 4, 8, 1, 2, 3, 4, 5, 9, 1, 2, 3, 4, 5, 6, 10, 1, 2, 3, 5, 7, 11, 1, 2, 3, 4, 6, 12, 1, 2, 3, 5, 6, 7, 13, 1, 2, 3, 6, 7, 14, 1, 3, 7, 15, 1, 2, 4, 8, 16, 1, 2, 3, 4, 5, 8, 9, 17, 1, 2, 3, 4, 5, 6

Offset

1,3

Comments

This sequence has similarities with A119709 and A165416; there we consider consecutive digits, here not.

Links

Rémy Sigrist, Rows n = 1..500 of triangle, flattened

Index entries for sequences related to binary expansion of n

Formula

T(n, 1) = 1.

T(n, A301977(n)) = n.

T(2^n, k) = 2^(k-1) for any n > 0 and k = 1..n+1.

T(2^n - 1, k) = 2^k - 1 for any n > 0 and k = 1..n.

Example

Triangle begins:
   1:    [1]
   2:    [1, 2]
   3:    [1, 3]
   4:    [1, 2, 4]
   5:    [1, 2, 3, 5]
   6:    [1, 2, 3, 6]
   7:    [1, 3, 7]
   8:    [1, 2, 4, 8]
   9:    [1, 2, 3, 4, 5, 9]
  10:    [1, 2, 3, 4, 5, 6, 10]
  11:    [1, 2, 3, 5, 7, 11]
  12:    [1, 2, 3, 4, 6, 12]
  13:    [1, 2, 3, 5, 6, 7, 13]
  14:    [1, 2, 3, 6, 7, 14]
  15:    [1, 3, 7, 15]
  16:    [1, 2, 4, 8, 16]

Maple

b:= proc(n) option remember; `if`(n=0, {0},
      map(x-> [x, 2*x+r][], b(iquo(n, 2, 'r'))))
    end:
T:= n-> sort([(b(n) minus {0})[]])[]:
seq(T(n), n=1..20);  # Alois P. Heinz, Jan 26 2022

Prog

(PARI) T(n, k) = my (b=binary(n), s=Set(1)); for (i=2, #b, s = setunion(s, Set(apply(v -> 2*v+b[i], s)))); return (s[k])

Crossrefs

Cf. A119709, A165416, A301977 (row length).

Sequence in context: A370329 A304038 A366113 * A165416 A222818 A057059

Adjacent sequences: A301980 A301981 A301982 * A301984 A301985 A301986

Keyword

nonn,base,tabf

Author

Rémy Sigrist, Mar 30 2018

Status

approved