Applications of Fibonacci Numbers: Volume 4 Proceedings of by Peter G. Anderson (auth.), G. E. Bergum, A. N. Philippou, A.

By Peter G. Anderson (auth.), G. E. Bergum, A. N. Philippou, A. F. Horadam (eds.)

This ebook includes thirty-three papers from one of the thirty-eight papers awarded on the Fourth foreign convention on Fibonacci Numbers and Their purposes which was once held at Wake wooded area college, Winston-Salem, North Carolina from July 30 to August three, 1990. those papers were chosen after a cautious evaluate via popular referees within the box, and so they diversity from common quantity thought to chance and information. The Fibonacci numbers and recurrence kinfolk are their unifying bond. it really is expected that this publication, like its 3 predecessors, can be valuable to analyze employees and graduate scholars drawn to the Fibonacci numbers and their purposes. March 1, 1991 The Editors Gerald E. Bergum South Dakota nation college Brookings, South Dakota, U. S. A. Alwyn F. Horadam college of latest England Armidale, N. S. W. , Australia Andreas N. Philippou Minister of schooling Ministry of schooling Nicosia, Cyprus xv THE ORGANIZING COMMITTEES neighborhood COMMITTEE foreign COMMITTEE Howard, Fred T. , Co-Chair Horadam, A. F. (Australia), Co-Chair Waddill, Marcellus E. , Co-Chair Philippou, A. N. (Cyprus), Co-Chair Hayashi, Elmer okay. Ando, S. (Japan) Bergum, G. E. (U. S. A. ) Vaughan, Theresa Harrell, Deborah Bicknell-Johnson, M. B. (U. S. A. ) Campbell, Colin (Scotland) Filipponi, Piero (Italy) Kiss, P. (Hungary) Turner, J. C. (New Zealand) xvii checklist OF individuals TO THE convention *ALFORD, CECIL zero. , (coauthor Daniel C. Fielder) "Pascal's Triangle: most sensible Gun or simply one of many Gang?" *ANDERSON, PETER G. , "A Fibonacci-Based Pseudo-Random quantity Generator.

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Theorem 1: Let b, b', and r satisfy the conditions of Lemma 1. Then there exists a one-to-one period-preserving map f between the set of all recurrences u(a, b) modulo p and the set of all recurrences u( a', b') modulo p, given by f(u(a,b)) =u(ra,b'), where a varies over all the residues modulo p, 2. PRELIMINARIES Associated with the Lucas sequence u( a, b) is the characteristic polynomial x 2 -ax-b with characteristic roots r l and r 2, Also associated with u( a, b) is the companion matrix _(0b a1) ' M- Note that the determinant of M is - b and the eigenvalues of M are the characteristic roots r l and r2, Given the vector (un' Un + 1)' it follows from the recursion relation defining u( a, b) that (2) Applying (2) n times to the vector (uo' u t ) = (0, 1), one sees that (3) It further follows by induction that (4) It is known (see [4, page 620]) and follows from (4) that Jl(a,b) = ord(M), (5) 39 PERIOD PATTERNS OF CERTAIN SECOND-ORDER ...

We have for the 512 sums E8 ;=1 . E(k, s, I, c) 2 = 1,640,100, and 8. 2 E E(I, s, r, c) = 1,640,100, ;=1 This same doubly-magic sum holds for the major space diagonals with these new three digit entries. BROKEN DIAGONALS We now explain the uppercase and the lowercase respectively over the columns A and C throughout the 4-Cube. Let I(n, 1, k, s), under some column labeled I, represent the number ADA, on the particular row where it is found, and as well let I(n, 2, k, s) represent DAD, where the indices k and s correspond to cube and square and n = 0, 1, ...

F. E. Bergum, Kluwer Academic Publishers, (1988) pp. 61-67. [2) Ando, S. and Sato, D. N. F. E. Bergum, Kluwer Academic Publishers, (1990) pp. 7-14. [3) Ando, S. and Sato, D. N. F. E. Bergum, Kluwer Academic Publishers, (1990) pp. 15-26. [4) Hillman, A. P. and Hoggatt, V. E. Jr. " The Fibonacci Quarterly, 10 (1972): pp. 565-568, 598. [5) Hitotumatu, S. and Sato, D. " The Fibonacci Quarterly [6) Hoggatt, V. E. Jr. and Alexanderson, G. L. " 13 (1975): p. 70. The Fibonacci Quarterly, 9 (1971): pp. 351-356, 420-421.

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