A136175 -id:A136175 - cp's OEIS frontend

A214830 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 8.

1, 8, 8, 17, 33, 58, 108, 199, 365, 672, 1236, 2273, 4181, 7690, 14144, 26015, 47849, 88008, 161872, 297729, 547609, 1007210, 1852548, 3407367, 6267125, 11527040, 21201532, 38995697, 71724269, 131921498, 242641464, 446287231, 820850193, 1509778888

Offset: 0

Abel Amene, Aug 07 2012

See comments in A214727.

Robert Price, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

a:=[1,8,8];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 24 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+7*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 24 2019

CoefficientList[Series[(x^2-7*x-1)/(x^3+x^2+x-1), {x, 0, 40}], x] (* Wesley Ivan Hurt, Jun 18 2014 *)
LinearRecurrence[{1,1,1}, {1,8,8}, 40] (* G. C. Greubel, Apr 24 2019 *)

my(x='x+O('x^40)); Vec((1+7*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 24 2019

((1+7*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 24 2019

G.f.: (1+7*x-x^2)/(1-x-x^2-x^3).

a(n) = -A000073(n) + 7*A000073(n+1) + A000073(n+2). - G. C. Greubel, Apr 24 2019

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.

A003265 Not representable by truncated tribonacci sequence 2, 4, 7, 13, 24, 44, 81, ....

Original entry on oeis.org

1, 3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 32, 34, 36, 38, 40, 42, 45, 47, 49, 52, 54, 56, 58, 60, 62, 65, 67, 69, 71, 73, 76, 78, 80, 82, 84, 86, 89, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 113, 115, 117, 119, 121, 123

Offset: 1

N. J. A. Sloane

nonn

The usual tribonacci representation of n writes n as a sum of tribonacci numbers 1, 2, 4, 7, 13, 24, ... (A000073) avoiding using three consecutive numbers (see A003726, A278038). But if we are not allowed to use 1, then some numbers cannot be represented, and such numbers are listed here. - N. J. A. Sloane, Oct 08 2018
Indices of odd terms of A003726. - Charlie Neder, Apr 25 2019
Numbers whose tribonacci representation ends in 1. Equivalently, the first column of the Trithoff (tribonacci) array, see A136175. - Tanya Khovanova and PRIMES STEP Senior group, May 07 2022

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charlie Neder, Table of n, a(n) for n = 1..1000
A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 65.

Cf. A000073, A003726, A136175, A278038.

Edited by N. J. A. Sloane, Oct 08 2018

a(47)-a(57) from Charlie Neder, Apr 25 2019

A353086 Column -1 of the extended Trithoff (Tribonacci) array.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 32, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 45, 46, 47, 47, 48, 49, 49, 50

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, Apr 22 2022

nonn

This column is also called the seed of the Trithoff array.
This is a nondecreasing sequence containing all nonnegative integers, with some of them doubled. The doubled numbers are the positions of the letter "a" in the Tribonacci word (A003144). Correspondingly, single numbers are positions of letters "b" and "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). The seed in the same row as n_1, n_2, and n_3 is 2n_2-n_3.

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

A351685 a(n) is the row of the Trithoff (tribonacci) array that contains the tails of the sequence which is n times the tribonacci numbers.

Original entry on oeis.org

1, 7, 10, 81, 101, 121, 141, 161, 1126, 1251, 1376, 1501, 1626, 1751, 1876, 2001, 2126, 2251, 2376, 2501, 2626, 17117, 17895, 18673, 19451, 20229, 21007, 21785, 22563, 23341, 24119, 24897, 25675, 26453, 27231, 28009, 28787, 29565, 30343, 31121, 31899, 32677, 33455

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 08 2022

nonn

This sequence is increasing.

The first term of the row a(n) in the Trithoff array is sequence A351689.

			Consider twice the tribonacci numbers: 0, 0, 2, 2, 4, 8, 14, 26, 48, and so on. The first few terms can be found in the first row of the Trithoff array A136175. The tail starting with 14, 26, and 48 is the seventh row of the Trithoff array. Thus, a(2) = 7.

Cf. A000073, A136175, A351689.

A351689 a(n) is the number in the first column of the Trithoff (tribonacci) array that starts off the row containing the tail of n times the tribonacci sequence.

Original entry on oeis.org

1, 14, 21, 176, 220, 264, 308, 352, 2466, 2740, 3014, 3288, 3562, 3836, 4110, 4384, 4658, 4932, 5206, 5480, 5754, 37510, 39215, 40920, 42625, 44330, 46035, 47740, 49445, 51150, 52855, 54560, 56265, 57970, 59675, 61380, 63085, 64790, 66495, 68200, 69905, 71610

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 08 2022

nonn

The sequence is increasing.
a(n) is divisible by n.
The row number where the tail of n times tribonacci numbers appears in the Trithoff array form sequence A351685.

			Consider twice the tribonacci numbers: 0, 0, 2, 2, 4, 8, 14, 26, 48, and so on. The first few terms can be found in the first row of the Trithoff array A136175. The tail starting with 14, 26, and 48 is the seventh row of the Trithoff array. The first number in the seventh row is 14. Thus, a(2) = 14.

Cf. A000073, A136175, A351685.

A353178 The row numbers of the Trithoff (tribonacci) array that correspond to difference sequences of other rows of the Trithoff array.

Original entry on oeis.org

2, 3, 4, 7, 11, 12, 16, 17, 19, 20, 21, 25, 26, 28, 29, 30, 33, 34, 35, 38, 42, 43, 45, 46, 47, 50, 51, 52, 55, 59, 60, 61, 64, 68, 69, 73, 74, 76, 77, 78, 81, 82, 83, 86, 90, 91, 92, 95, 99, 100, 104, 105, 107, 108, 109, 112, 116, 117, 121, 122, 124, 125, 126, 130, 131

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, Apr 28 2022

nonn

All positive tribonacci-like sequences are in the Trithoff array.
Every tribonacci-like sequence s is a difference sequence of another tribonacci-like sequence t, where t is uniquely defined. If s is an integer sequence then, t doesn't have to be an integer sequence. If t is an integer sequence, then the row number corresponding to sequence s is in this sequence.
These are the Trithoff array rows that are all odd, or all even, or alternate between even and odd.

			The first row of the Trithoff array is the sequence of tribonacci numbers A000073. Its differences form sequence A001590, which is the second row of the Trithoff array. Thus, 2 is in this sequence.
The tribonacci sequence, the first row of the Trithoff array, is the difference sequence of the tribonacci-like sequence A000213 divided by 2. The result is not an integer sequence. Thus, 1 is not in this sequence.

Complement of A353193.

Cf. A000073, A000213, A001590, A136175.

A353193 The row numbers of the Trithoff (tribonacci) array that don't correspond to difference sequences of other rows of the Trithoff array.

Original entry on oeis.org

1, 5, 6, 8, 9, 10, 13, 14, 15, 18, 22, 23, 24, 27, 31, 32, 36, 37, 39, 40, 41, 44, 48, 49, 53, 54, 56, 57, 58, 62, 63, 65, 66, 67, 70, 71, 72, 75, 79, 80, 84, 85, 87, 88, 89, 93, 94, 96, 97, 98, 101, 102, 103, 106, 110, 111, 113, 114, 115, 118, 119, 120, 123, 127, 128, 129, 132, 136, 137, 141, 142, 144

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, Apr 29 2022

nonn

All positive tribonacci-like sequences are in the Trithoff array.
Every tribonacci-like sequence s is a difference sequence of another tribonacci-like sequence t, where t is uniquely defined. If s is an integer sequence then, t doesn't have to be an integer sequence. If t is an integer sequence, then the row number corresponding to sequence s is not in this sequence.
These are the Trithoff array rows that repeat a pattern: even, even, odd, odd.

			The first row of the Trithoff array is the sequence of tribonacci numbers A000073. Its differences form sequence A001590, which is the second row of the Trithoff array. Thus, 2 is not in this sequence.
The tribonacci sequence, the first row of the Trithoff array, is the difference sequence of the tribonacci-like sequence A000213 divided by 2. The result is not an integer sequence. Thus, 1 is in this sequence.

Complement of A353178.

Cf. A000073, A000213, A001590, A136175.

cp's OEIS Frontend

A214830 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 8.

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

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A003265 Not representable by truncated tribonacci sequence 2, 4, 7, 13, 24, 44, 81, ....

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A353086 Column -1 of the extended 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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A351685 a(n) is the row of the Trithoff (tribonacci) array that contains the tails of the sequence which is n times the tribonacci numbers.

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A351689 a(n) is the number in the first column of the Trithoff (tribonacci) array that starts off the row containing the tail of n times the tribonacci sequence.

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A353178 The row numbers of the Trithoff (tribonacci) array that correspond to difference sequences of other rows of the Trithoff array.

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A353193 The row numbers of the Trithoff (tribonacci) array that don't correspond to difference sequences of other rows of the Trithoff array.

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