A356784 Inventory of positions as an irregular table; row 0 contains 0, subsequent rows contain the 0-based positions of 0's, followed by the position of 1's, of 2's, etc. in prior rows flattened.

0, 0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 4, 3, 5, 6, 7, 0, 1, 2, 4, 8, 3, 5, 9, 6, 10, 7, 12, 11, 13, 14, 15, 0, 1, 2, 4, 8, 16, 3, 5, 9, 17, 6, 10, 18, 7, 12, 21, 11, 19, 13, 22, 14, 24, 15, 26, 20, 23, 25, 28, 27, 29, 30, 31, 0, 1, 2, 4, 8, 16, 32, 3, 5, 9, 17, 33

Offset

0,7

Comments

The n-th row contains A011782(n) terms, and is a permutation of 0..A011782(n)-1.

The leading term of each row is 0, and is followed by powers of 2, and then by positive nonpowers of 2.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8191

Rémy Sigrist, Scatterplot of first 2^22 terms

Rémy Sigrist, C++ program

Formula

a(n) = 0 iff n belongs to A131577.

a(n) = 1 iff n belongs to A000051 \ {2}.

a(n) = 2 iff n belongs to A052548 \ {3, 4}.

a(n) = 3 iff n belongs to A005126 \ {2, 4}.

T(n, 0) = 0.

T(n, k) = 2^(k-1) for k = 1..n-1.

Example

Table begins:
   0,
   0,
   0, 1,
   0, 1, 2, 3,
   0, 1, 2, 4, 3, 5, 6, 7,
   0, 1, 2, 4, 8, 3, 5, 9, 6, 10, 7, 12, 11, 13, 14, 15,
   ...
For n = 5:
- the terms in rows 0..4 are: 0, 0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 4, 3, 5, 6, 7,
- we have 0's at positions 0, 1, 2, 4, 8,
- we have 1's at positions 3, 5, 9,
- we have 2's at positions 6, 10,
- we have 3's at positions 7, 12,
- we have one 4 at position 11,
- we have one 5 at position 13,
- we have one 6 at position 14,
- we have one 7 at position 15,
- so row 5 is: 0, 1, 2, 4, 8, 3, 5, 9, 6, 10, 7, 12, 11, 13, 14, 15.

Prog

(C++) See Links section.
(Python)
terms = [0, ]
for i in range(1, 10):
    new_terms = []
    for j in range(max(terms)+1):
        for k in range(len(terms)):
            if terms[k] == j: new_terms.append(k)
    terms.extend(new_terms)
print(terms) # Gleb Ivanov, Nov 01 2022

Crossrefs

Cf. A000051, A005126, A011782 (row lengths), A052548, A131577, A342585, A357317, A357491.

Sequence in context: A362190 A274650 A294648 * A359941 A360302 A351939

Adjacent sequences: A356781 A356782 A356783 * A356785 A356786 A356787

Keyword

nonn,tabf

Author

Rémy Sigrist, Oct 01 2022

Status

approved