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

Author

Rémy Sigrist, Oct 01 2022

Keywords

• nonn
• tabf

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.

Examples

			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.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8191
• Rémy Sigrist, Scatterplot of first 2^22 terms
• Rémy Sigrist, C++ program

Crossrefs

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

Programs

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

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.