A353193 -id:A353193 - cp's OEIS frontend

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

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.

A354215 a(n) is the row number of the Trithoff (tribonacci) array where we can find the tail of the following sequence: apply the difference operator n times to the tribonacci sequence.

Original entry on oeis.org

1, 2, 3, 7, 19, 29, 81, 125, 353, 161, 1545, 705, 2001, 3089, 8769, 24897, 38433, 109121, 309825, 478273, 1357953, 2096257, 5951873, 2715905

Offset: 0

Tanya Khovanova and PRIMES STEP Senior group, May 19 2022

The tribonacci sequence has a repeated pattern even, even, odd, odd. Its difference sequence alternates between even and odd. The second difference sequence consists only of odd numbers. The third or higher difference sequence consists only of even numbers. It follows that rows a(n) in the Trithoff array, for n > 2, contain all even numbers.

			Consider the tribonacci sequence A000073: 0, 0, 1, 1, 2, 4, 7, 13, .... Its first difference sequence is sequence A001590: 0, 1, 0, 1, 2, 3, 6, ... This sequence follows the tribonacci rule and its tail starting from number 3 is the second row of the Trithoff array A136175. Thus, a(1) = 2.

Cf. A000073, A136175, A351685, A351689, A353178, A353193.

cp's OEIS Frontend

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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