On the arrowhead-Fibonacci numbers

Abstract In this paper, we define the arrowhead-Fibonacci numbers by using the arrowhead matrix of the characteristic polynomial of the k-step Fibonacci sequence and then we give some of their properties. Also, we study the arrowhead-Fibonacci sequence modulo m and we obtain the cyclic groups from the generating matrix of the arrowhead-Fibonacci numbers when read modulo m. Then we derive the relationships between the orders of the cyclic groups obtained and the periods of the arrowhead-Fibonacci sequence modulo m.


Introduction
It is well-known that a square matrix is called an arrowhead matrix if it contain zeros all in entries except for the first row, first column, and main diagonal. In other words, an arrowhead matrix M D m i;j .n/ .n/ is defined as follows: .
The k-step Fibonacci sequence˚F k n « is defined recursively by the equation For detailed information about the k-step Fibonacci sequence, see [1,2]. It is clear that the characteristic polynomial of the k-step Fibonacci sequence is as follows: Suppose that the .n C k/th term of a sequence is defined recursively by a linear combination of the preceding k terms: a nCk D c 0 a n C c 1 a nC1 C C c k 1 a nCk 1 where c 0 ; c 1 ; : : : ; c k 1 are real constants. In [1], Kalman derived a number of closed-form formulas for the generalized sequence by the companion matrix method as follows: Let the matrix A be defined by A D a i;j k k D for n > 0.
Many of the obtained numbers by using homogeneous linear recurrence relations and their miscellaneous properties have been studied by many authors; see, for example, [3][4][5][6][7][8][9][10][11]. Arrowhead-Fibonacci numbers for the 2-step Pell and Pell-Lucas sequences were illustrated in [12]. In Section 2, we define the arrowhead-Fibonacci numbers by using the arrowhead matrix N , which is defined by the aid of the characteristic polynomial of the k-step Fibonacci sequence. Then we derive their miscellaneous properties such as the generating matrix, the combinatorial representation, the Binet formula, the permanental representations, the exponential representation and the sums.
The study of recurrence sequences in groups began with the earlier work of Wall [13], where the ordinary Fibonacci sequence in cyclic groups were investigated. The concept extended to some special linear recurrence sequences by some authors; see, for example, [3,[14][15][16]. In [3,15,17], the authors obtained the cyclic groups via some special matrices. In Section 3, we study the arrowhead-Fibonacci sequence modulo m: Also in this section, we obtain the cyclic groups from the multiplicative orders of the generating matrix of the arrowhead-Fibonacci sequence such that the elements of the generating matrix when read modulo m. Then we obtain the rules for the orders of the obtained cyclic groups and we give the relationships between the orders of those cyclic groups and the periods of the arrowhead-Fibonacci sequence modulo m.

The arrowhead-Fibonacci numbers
We next define the arrowhead matrix N D n i;j .kC1/ .kC1/ by using the characteristic polynomial of the k-step Fibonacci sequence P F k .x/ as follows: . Now we consider a new .k C 1/-step sequence which is defined by using the matrix N and is called the arrowhead-Fibonacci sequence. The sequence is defined by integer constants a kC1 .1/ D D a kC1 .k/ D 0 and a kC1 .k C 1/ D 1 and the recurrence relation a kC1 .n C k C 1/ D a kC1 .n C k/ a kC1 .n C k 1/ a kC1 .n/ for n 1, where k 2. By (1), we can write a generating matrix for the arrowhead-Fibonacci numbers as follows: ; where n k, a kC1 .u/ is denoted by a u kC1 and G 0 kC1 is a .k C 1/ .k 1/ matrix as follows: : : : a n kC1 C a n 1 kC1 C C a n kC1 kC1 Á a n kC1 C a n 1 kC1 C a n 2 kC1 Á a n kC1 C a n 1 It is important to note that det G kC1 D . 1/ kC1 and the Simpson identity for a recursive sequence can be obtained from the determinant of its generating matrix. From this point of view, we can easily derive the Simpson formulas of the arrowhead-Fibonacci sequences for every k 2.

Theorem 2.3 (Chen and Louck
where the summation is over nonnegative integers satisfying is a multinomial coefficient, and the coefficients in (3) are defined to be 1 if u D i j .
Then we can give a combinatorial representation for the arrowhead-Fibonacci numbers by the following Corollary.
where the summation is over nonnegative integers satisfying t 1 C 2t 2 C C .k C 1/ t kC1 D n: , then the proof is immediately seen from .G kC1 / n . Now we consider the Binet formulas for the arrowhead-Fibonacci numbers by using the determinantal representation.
It is clear that f .0/ D 1 and f .1/ D k for all k 2. Let u be a multiple root of f .x/, then u … f0; 1g. If possible, u is a multiple root of f .x/ in which case f .u/ D 0 and f 0 .u/ D 0. Now f 0 .u/ D 0 and u ¤ 0 give C.kC2/ 2 which are contradictions since k 2. If x 1 , x 2 , : : :, x kC1 are roots of the equation x kC1 x k C x k 1 C x k 2 C C 1 D 0, then by Lemma 2.5, it is known that x 1 , x 2 , : : :, x kC1 are distinct. Let V kC1 be .k C 1/ .k C 1/ Vandermonde matrix as follows: x kC1 / k 1 : : : : : : : : : Proof. Since the eigenvalues of the matrix G kC1 , x 1 , x 2 , : : :, x kC1 are distinct, the matrix G kC1 is diagonalizable. Let D kC1 D .x 1 ; x 2 ; : : : ; x kC1 /, we easily see that Then we can write the following linear system of equations: x 2 / nCkC1 i : : : for n k 2. So, for each i; j D 1; 2; : : : ; k C 1, we obtain g .n/ i;j as follows Then we can give the Binet formulas for the arrowhead-Fibonacci numbers by the following corollary.
Corollary 2.7. Let a kC1 .n/ be the nth the arrowhead-Fibonacci number for k 2. Then Now we consider the permanent representations of the arrowhead-Fibonacci numbers.
Definition 2.8. A u v real matrix M D m i;j is called a contractible matrix in the k th column (resp. row) if the k th column (resp. row) contains exactly two non-zero entries.
Suppose that x 1 ; x 2 ; : : : ; x u are row vectors of the matrix M . If M is contractible in the k th column such that m i;k ¤ 0,m j;k ¤ 0 and i ¤ j , then the .u 1/ .v 1/ matrix M ij Wk obtained from M by replacing the i th row with m i;k x j C m j;k x i and deleting the j th row. The k th column is called the contraction in the k th column relative to the i th row and the j th row.
In [21], Brualdi  It is easy to see that perM kC1 .2/ D 0 and perM kC1 .3/ D 2. From definition of arrowhead-Fibonacci sequence it is clear that a kC1 .k C 2/ D 1, a kC1 .k C 3/ D 1 and a kC1 .k C 4/ D 2. So we have the conclusion for n < 4. Let the equation hold for n > 4, then we show that the equation holds for n C 1. If we expand the perM kC1 .n/ by the Laplace expansion of permanent with respect to the first row, then we obtain perM kC1 .n C 1/ D perM kC1 .n/ perM kC1 .n 1/ perM kC1 .n k/ .
Then we can give more general results by using other permanent representations than the above.
(ii) For n > k C 2, Proof. (i). Let the equation hold for n > k C 1, then we show that the equation holds for n C 1. If we expand the perR kC1 .n/ by the Laplace expansion of permanent with respect to the first row, then we obtain perR kC1 .n C 1/ D perR kC1 .n/ perR kC1 .n 1/ perR kC1 .n k/ .
(ii) It is clear that expanding the perT kC1 .n/ by the Laplace expansion of permanent with respect to the first row, gives us perT kC1 .n/ D perT kC1 .n 1/ C perR kC1 .n/ .
Then, by the result of Theorem 2.10. (i) and an induction on n, the conclusion is easily seen.
It is easy to show that the generating function of the arrowhead-Fibonacci sequence fa kC1 .n/g is as follows: Then we can give an exponential representation for the arrowhead-Fibonacci numbers by the aid of the generating function with the following theorem.
Theorem 2.11. The arrowhead-Fibonacci numbers have the following exponential representation: where k 2. Proof.
Thus we have the conclusion.  Proof. Let X D f.x 1 ; x 2 ; : : : ; x kC1 / j x i 's be integers such that 0 Ä x i Ä m 1g. Since there are m kC1 distinct .k C 1/-tuples of elements of Z m , at least one of the .k C 1/-tuples appears twice in the sequence n a m kC1 .n/ o . Thus, the subsequence following this .k C 1/-tuple repeats; hence, the sequence is periodic. Assume that u > v and From the definition of the arrowhead-Fibonacci sequence, we can easily derive a m kC1 .n/ D a m kC1 .n C k C 1/ C a m kC1 .n C k/ a m kC1 .n C k 1/ a m kC1 .n C 1/ Given an integer matrix X D x i;j , X .mod m/ means that all entries of X are modulo m, that is, X .mod m/ D x i;j .mod m/ . Let us consider the set hXi m D˚X i .mod m/ j i 0 « . If gcd .m; det X / D 1, then hX i m is a cyclic group. Let jhX i m j denote the cardinality of the set hXi m . Since det G kC1 D . 1/ kC1 , hG kC1 i m is a cyclic group for every positive integer m. By (2), it is easy to show that P kC1 .m/ DˇhG kC1 i mˇf or k 2.
Now we give some useful properties for the period P kC1 .m/ by the following theorem. (i) Let p be a prime and suppose that u is the smallest positive integer with P kC1 p uC1 ¤ P kC1 .p u /. Then P kC1 .p v / D p v u P kC1 .p/ for every v > u and k 2. .p i /˛i , .u > 1/, then P kC1 .m/ equals the least common multiple of the P kC1 .p i /˛i 's for k 2. (iii) If k is a even integer such that k 2, then P kC1 .m/ is even for every positive integer m.
Proof. (i) If I is the .k C 1/ .k C 1/ identity matrix and t is a positive integer such that .G kC1 / P kC1. p tC1 / Á I modp tC1 ;then .G kC1 / P kC1. p tC1 / Á I modp t . Then, it is clear that P kC1 p t divides P kC1 p tC1 . On the other hand, if we denote .G kC1 / P kC1. p t / D I C a .t / i;j p t Á , then by the binomial expansion, we may write This yields that P kC1 p t p is divisible by P kC1 p t . Then, P kC1 p t C1 D P kC1 p t or P kC1 p tC1 D P kC1 p t p, and the latter holds if and only if there is a a .t / i;j which is not divisible by p. Due to fact that we assume u is the smallest positive integer such that P kC1 p uC1 ¤ P kC1 .p u /, there is an a .u/ i;j which is not divisible by p. Since there is a a .u/ i;j such that p does not divide a .u/ i;j , it is easy to see that there is an a .uC1/ i;j which is not divisible by p. This shows that P kC1 p uC2 ¤ P kC1 p uC1 . Then we get that P kC1 p uC2 D p P kC1 p uC1 D p .p P kC1 .p u // D p 2 P kC1 .p u /. So by induction on u we obtain P kC1 .p v / D p v u P kC1 .p/ for every v > u. In particular, if P kC1 p 2 ¤ P kC1 .p/, then P kC1 .p v / D p v 1 P kC1 .p/.
(ii) It is clear that the sequence n a .p i /˛i kC1 .n/ o repeats only after blocks of length P kC1 .p i /˛i where is a natural number. Since P kC1 .m/ is period of the sequence n a m kC1 .n/ o ; the sequence n a .p i /˛i kC1 .n/ o repeats after P kC1 .m/ terms for all values i . Thus, we easily see that P kC1 .m/ is of the form P kC1 .p i /˛i for all values of i , and since any such number gives a period of P kC1 .m/. Therefore, we conclude that P kC1 .m/ D lcm P kC1 .p 1 /˛1 ; : : : ; P kC1 .p u /˛u .
(iii) Since det G kC1 D 1 when k is a even integer and P kC1 .m/ DˇhG kC1 i mˇ, we have the conclusion.