Smarandache cyclic geometric determinant sequences

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Scientia Magna Vol. 8 (2012), No. 4, 88-91

Smarandache cyclic geometric determinant sequences A. C. F. Bueno Department of Mathematics and Physics, Central Luzon State University, Science City of Mu˜ noz 3120, Nueva Ecija, Philippines Abstract In this paper, the concept of Smarandache cyclic geometric determinant sequence was introduced and a formula for its nth term was obtained using the concept of right and left circulant matrices. Keywords Smarandache cyclic geometric determinant sequence, determinant, right circulan -t matrix, left circulant matrix.

§1. Introduction and preliminaries Majumdar [1] gave the formula for nth term of the following sequences: Smarandache cyclic natural determinant sequence, Smarandache cyclic arithmetic determinant sequence, Smarandache bisymmetric natural determinant sequence and Smarandache bisymmetric arithmetic determinant sequence. Definition 1.1. A Smarandache cyclic geometric determinant sequence {SCGDS(n)} is a sequence of the form  ¯  ¯   ¯ a {SCGDS(n)} = |a|, ¯¯   ¯ ar 

¯ ¯ ¯ a ¯ ¯ ar ¯ ¯ ¯ , ¯ ar ¯ ¯ a ¯ ¯ ¯ ar2

Definition 1.2. A matrix RCIRCn (~c) ∈ Mnxn (R) is if it is of the form  c0 c1 c2 ...   cn−1 c0 c1 ...    cn−2 cn−1 c0 ...  RCIRCn (~c) =  . .. .. . .  .. . . .    c c3 c4 ...  2 c1 c2 c3 ... where ~c = (c0 , c1 , c2 , ..., cn−2 , cn−1 ) is the circulant vector.

ar ar2 a

¯   ar2 ¯¯   ¯ a ¯¯ , . . . .   ¯  ar ¯

said to be a right circulant matrix

cn−2 cn−3 cn−4 .. . c0 cn−1

cn−1

 cn−2    cn−3   , ..  .    c1   c0


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