Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)

MÁTYÁS, F., Genaralize d Fibonacci-type numbers as matrix determinants

Genaralized Fibonacc i -type numbers as matrix determinants 41 2. Result We shall prove the following theorem. Theorem. Let the squence {G

n}^L

2-fc be defined by (2), where p\G\ ^ 0

, pk = ±1 and k > 2. Let the matrix A

u x

n be defined by (•}). I hen for every n 1 G

n = det(Anxn). Remark. In the case k = 2 our matrices A

nx

n are of tridiagonal ones. Proof. First we consider the case 1 < n < k. Then, for n = 1 det(A

lx

l) = 6'i. If n = 2 or 3, then 1 , I Gi —e

3G2-k\ r, Q /1 det 3 = Pi GI -e G 2-k VP

1 J - Pi GI + PKG

2-K = G

2 and / ^ 4 s Í \ 3-k \ = PiG

2 - E

4G

3-FCF

6 — PiG-2 - e

2G

3-k = PIG

2 + PkG

3-k = G

3. Suppose that G

n_j = det(A

n_

;x

n_

;

) (j = 1,2,3) holds for an integer n, where 4 < n < A;. Then, developing the determinant i Gi -e

3G

2_

f

e —e

4G det e3 Pi 0 1 1 o -e

3 Pi det (A,

lX

n) = det ( Gi -e

3 0 „•if c CT 2 — A; Pi -e

3 0 Pi —e

nG

n-i-k 0 0 -e

n+

1G

n 0 0 \ 0 0 0 -e

3 Pi with respect to the last column, we have (let (A

n

Xn) — P\G

n- (-l)"+V

+

1G, / Q x n — 1 i-k(~e

3) - Pi G

n l)

2n+

1e

4

n ~ 2Gn-k — PlG„-i + PkGn-k = G

r

i

. That is, our theorem holds for every n

, if I < » <