Abstract
For polynomials and satisfying the noncommutative multiplication , let and be the arithmetic tables, respectively. We investigate sequential properties of various diagonal sums over the tables and and prove that they are types of interlocked Fibonacci sequence and Padovan sequence.
1. Introduction
The Pascal table is an arithmetic table of a polynomial , in which is assumed tactically. The Pauli Pascal table is an arithmetic table of with noncommuting variables such that . The and satisfy the following (see [1]): It is known that diagonal sums over give a Fibonacci sequence , while those over yield a sequence that is interlocked by two Fibonacci sequences [2].
Consider a polynomial with negative exponent satisfying either or and denote the corresponding arithmetic table by either or with , respectively.
By a -slope diagonal over a table (), we mean a generalized diagonal moving steps along -axis and steps along -axis. Over , let denote the -slope diagonal set starting from toward northeast direction, and be the sum of elements in . We call it the -slope th diagonal sum. Similarly, over , let be the -slope diagonal set starting from toward southwest direction, and be the -slope th diagonal sum. When , we simply say -slope diagonal sums and .
A purpose of the work is to study arithmetic tables and . We investigate sequential properties of generalized diagonal sums and and find their interrelationships. We particularly give attention to of and prove is a type of interlocked Padovan sequence. The results of the work provide interesting connections of sequences over the arithmetic tables of having either commutative or noncommutative rules.
2. Arithmetic Table and Its Diagonal Sum
The arithmetic table of with can be obtained by Taylor series expansion. Every element () of in Table 1 satisfies a recurrence rule .
By flipping and passing it over , the pile-up table follows the Pascal rule (1). For example, we get . Some recurrence rules of -slope diagonal sum over and -slope diagonal sum over were studied as follows.
Lemma 1 (see [3, 4]). with initials . And with initials . Moreover, () with initials .
The proof of Lemma 1 is mainly due toLet and be subsequences having only eventh and oddth terms, respectively, in . From Table 2, we observeSince both and are Fibonacci sequences, we say is an interlocked Fibonacci sequence. For any , a sequence satisfying with initials is called a Fibo -sequence [5].
Theorem 1. is an interlocked Fibo -sequence for .
Proof. Let and be subsequences consisting of eventh or oddth terms, respectively, in . When , is clearly an interlocked Fibo 1-sequence. When , Table 2 shows thatwhere and are Fibo 2-sequences with initials and . Similarly, shows and are Fibo 3-sequences having initials and . So, with is an interlocked Fibo -sequence.
In general, for any , Lemma 1 shows that the sequenceholds () with initials . Thus, and satisfy recurrences and . So, they are Fibo -sequences having initials and , respectively. Hence, is an interlocked Fibo -sequence.
Now, in order to have the table of with , look at the piled-up table satisfying the Pauli rule (1).
Then, by flipping the upper part upside down, we get (Table 3) holdingThus, and yield expansions of with , for instance, .
Theorem 2. . In particular, and for any .
Proof. Since , we may assume for some .
If , then recurrence (6) impliesSimilarly, if , thenThe other cases can be proved analogously. In particular, , so .
Theorem 2 can be compared to in (2).
3. Diagonal Sum over
We will discuss -slope diagonal and its sum on .
Theorem 3. When , , , and for .
When , for .
Proof. The first few 1-slope diagonal sets and sums in are as follows:In general, by (6) and for all (Theorem 2), we havebecause , and in Theorem 2.
Similarly, and also showTherefore, we have .
The -slope diagonals sets and their sums in are as follows:Thus, satisfiesSo, we have for .
Now, when , we havefor . Hence, .
On the other hand, when , we havefor . Thus, .
Let us continue to work with -slope diagonals in .
Theorem 4. For , .
Proof. Observe and fromLet . Since and for , , we haveNow, let . Again with (, ), we also haveWe now have recurrence rules of -slope diagonal sums over .
Theorem 5. with even .
And with odd .
Proof. Let . Since , recurrence (6) yieldsNow, assume and . Then, for . However, since , we haveSimilarly, if and , then , soA more explicit relation of the diagonal sets and is as follows.
Theorem 6. Let and be the th elements of the sets and , respectively. Then, and . In general, any th elements in and in are the same, except for signs.
Proof. Note of and of . Theorem 2 and the symmetricity of implywhich shows for .
Now, for any th element in the diagonal set , we note thatBy mod 4, if , then or 2 according to or . Thus, implies or 3 according to or . Similarly, means (if ) or (if ). And says (if ) or (if ).
Thus, we have the following 4 cases (all congruences are by mod 4):(i)Let . If then , so . If then , so . Thus, for any .(ii)Let . If then , so . If then , so . Thus, for any .(iii)Let . If then , so . If then , so .(iv)Let . If , then , so . If , then , so .Therefore, we have .
Similarly, in the set , the th element () arewhere all congruences are by mod 4. Thus, we generally haveNow, the next table shows and for the first few .And in general by Theorem 2, we have (congruences are by mod 4)We are now ready to obtain a recurrence on .
Theorem 7. for any .
Proof. Theorem 3 implies and . So, we haveSimilarly, Theorem 4 shows and . Thus, if , thenwhile if , thenNow, consider any . If , then and by Theorem 5. So,On the other hand, if , then and . However, since the latter identity equals , and the sum of the two identities yields .
4. Extended Sequences of and
Lemma 1 shows that is a Fibo -sequence. By extending subscripts backward up to all integers, we have a sequence satisfying , which is also a Fibo -sequence. On the contrary, the sequence satisfies . By extending to all integers, we get a sequence in [3] satisfying
In fact, from , we havesuch that (see Table 4).
A sequence satisfying with initials is called a Padovan -sequence [6, 7]. In particular, it is a Fibonacci sequence if . Identity (32) yields the next lemma immediately.
Lemma 2 (see [3]). is a Padovan -sequence with initials and .
Now, over and , we consider extended sequences of and , in which subscripts are extended to all integers. From in Lemma 1, we easily have an extended sequence of satisfying
So, is an interlocked Fibo -sequence as in Theorem 1.
On the contrary, let for . Then, in Theorem 7 implies . So, by setting , we have
That is, . It shows that is an extended sequence of satisfying
Theorem 8. is an interlocked Padovan -sequence with initials and .
Proof. The subsequence having only eventh terms of isThen, in (36) implies that the sum of consecutive two eventh terms equals distanced eventh term. So, is a Padovan -sequence. Similarly, the subsequence of oddth terms of also satisfies that the sum of consecutive two oddth terms equals distanced oddth term. Thus, is an interlocked Padovan -sequence.
Corollary 1. The sequences and are equal to and , respectively. And is an interlocked Fibo 5-sequence.
Proof. The proof is due to Tables 4 and 5. And the interlocked Padovan 3-sequence in Theorem 8 satisfies . Thus,which is a recurrence of interlocked Fibo 5-sequence.
We note that is interlocked by two Padovan subsequenceswith initials . However, is interlocked by distinct Padovan 4-sequenceswhere these two sequences satisfy . Moreover, the sequences are obtained explicitly from Pascal and Pauli tables as follows. Write and by means of th rows and . Letbe tables having duplicated rows of and .
Theorem 9. In , the sequence of 1-slope diagonal sums equals . And in , the sequence of 1-slope diagonal sums is interlocked by two Padovan 3-sequences and , where is the ordinary Padovan sequence such that for all .
Proof. From , the sequence of 1-slope diagonal sums is that corresponds to . Similarly, the sequence of 1-slope diagonal sums of isThat satisfies for some . NoteThen, for all by the Pauli recurrence (1). And is an interlocked Padovan 3-sequence by of eventh terms and of oddth terms. Clearly, is the ordinary Padovan sequence satisfying for all .
Data Availability
The data used to support the findings of the study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.