A353086 -id:A353086 - cp's OEIS frontend

A353084 Column 0 of the extended Trithoff (tribonacci) array.

1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 80

Offset: 1

Tanya Khovanova and PRIMES STEP Senior Group, Apr 22 2022

nonn

This column is also called the wall of the Trithoff array.
These are the positions of letters a and b in the tribonacci word.
Complement of A003146: position of letter c in the tribonacci word.
Suppose number n_1 has tribonacci representation t that ends in 1 (such numbers are in column 1 of the Trithoff array). Then its tribonacci successor n_2 has tribonacci representation t0 (such numbers are in column 2 of the Trithoff array), and the successor of the successor n_3 has tribonacci representation t00 (such numbers are in column 3 of the Trithoff array). This sequence consists of numbers n_3-n_2-n_1.

			The first few tribonacci numbers are 1, 2, 4, 7, 13, 24, 44. The number 23 can be represented as 13+7+2+1. Thus, its tribonacci representation is 11011. The tribonacci successor of 23 is 24+13+4+2 = 43, and the next successor is 44+24+7+4 = 79. Thus, 79 - 43 - 23 = 13 is in this sequence.

Cf. A000073, A003144, A003145, A003146, A136175, A353083, A353086, A353090.

A353083 The second column of the Trithoff (tribonacci) array.

Original entry on oeis.org

2, 6, 9, 15, 19, 22, 26, 30, 33, 39, 43, 46, 50, 53, 59, 63, 66, 70, 74, 77, 83, 87, 90, 96, 100, 103, 107, 111, 114, 120, 124, 127, 131, 134, 140, 144, 147, 151, 155, 158, 164, 168, 171, 175, 179, 182, 188, 192, 195, 199, 202, 208, 212, 215, 219, 223, 226

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, Apr 22 2022

nonn

These are the numbers whose tribonacci representation ends in 10.

This is a subsequence of A003145: Numbers in this sequence indicate positions of letter b in the tribonacci word, but not all such positions.

			The first few tribonacci numbers are 1, 2, 4, 7, 13, 24. The number 43 can be represented as 24+13+4+2. Thus, its tribonacci representation is 110110, and 43 is in this sequence.

Cf. A000073, A003145, A003265, A136175, A353084, A353086, A353090.

A353090 Column -2 of the extended Trithoff (tribonacci) array.

Original entry on oeis.org

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

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, Apr 22 2022

nonn

This column is also called the pre-seed of the Trithoff array.
This is a nondecreasing sequence containing all nonnegative integers, with some of them doubled and some tripled. The doubled numbers are the positions of the letter "c" in the tribonacci word (A003146). Correspondingly, the tripled numbers are positions of letters "a" and "b" in the tribonacci word.
Suppose number n_1 has tribonacci representation t that ends in 1 (such numbers are in column 1 of the Trithoff array). Then its tribonacci successor n_2 has tribonacci representation t0 (such numbers are in column 2 of the Trithoff array), and the successor of the successor n_3 has tribonacci representation t00 (such numbers are in column 3 of the Trithoff array). The pre-seed in the same row as n_1, n_2, and n_3 is 2n_1-n_2.

Cf. A000073, A003144, A003145, A003146, A136175, A353083, A353084, A353086.

A351631 The numbers that are not doubled in column -1 of the extended Trithoff (tribonacci) array.

Original entry on oeis.org

0, 2, 4, 6, 9, 11, 13, 15, 17, 19, 22, 24, 26, 28, 30, 33, 35, 37, 39, 41, 43, 46, 48, 50, 53, 55, 57, 59, 61, 63, 66, 68, 70, 72, 74, 77, 79, 81, 83, 85, 87, 90, 92, 94, 96, 98, 100, 103, 105, 107, 109, 111, 114, 116, 118, 120, 122, 124, 127, 129, 131, 134, 136, 138, 140, 142, 144, 147, 149, 151

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 04 2022

nonn

Excluding zeros, these are the indices of letters b and c in the tribonacci word.
The complement of A003144: the indices of the letter a in the tribonacci word. These numbers are doubled in column -1 in the extended Trithoff array.
Also integers with "even" lazy tribonacci representation A352103, and first column of A385436. - A.H.M. Smeets, Jun 29 2025

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A000073, A003144, A003145, A003146, A136175, A353083, A353084, A353086, A353090.

def ToDual_111_Zeck(n):
    if n == 0:
        return "0"
    f0, f1, f2, sf = 1, 0, 0, 0
    while n > sf:
        f0, f1, f2 = f0+f1+f2, f0, f1
        sf += f0
    r, s = sf-n, "1"
    while f0 > 1:
        f0, f1, f2 = f1, f2, f0-f1-f2
        r, s = r%f0, s+str(1-r//f0)
    return s
n, a = 0, 0
while n < 70:
    s = ToDual_111_Zeck(a)
    if s[len(s)-1] == "0": # == even
        n += 1
        print(a, end = ", ")
a += 1 # A.H.M. Smeets, Jun 28 2025

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A353084 Column 0 of the extended Trithoff (tribonacci) array.

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A353083 The second column of the Trithoff (tribonacci) array.

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A353090 Column -2 of the extended Trithoff (tribonacci) array.

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A351631 The numbers that are not doubled in column -1 of the extended Trithoff (tribonacci) array.

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