# Applications of Fibonacci Numbers: Volume 9: Proceedings of by Arnold Adelberg (auth.), Frederic T. Howard (eds.)

By Arnold Adelberg (auth.), Frederic T. Howard (eds.)

This publication includes 28 learn articles from one of the forty nine papers and abstracts provided on the 10th overseas convention on Fibonacci Numbers and Their functions. those articles were chosen after a cautious overview by means of professional referees, they usually variety over many parts of arithmetic. The Fibonacci numbers and recurrence kin are their unifying bond. We observe that the thing "Fibonacci, Vern and Dan" , which follows the creation to this quantity, isn't really a study paper. it's a own memory by way of Marjorie Bicknell-Johnson, an established member of the Fibonacci organization. The editor believes it will likely be of curiosity to all readers. it truly is expected that this booklet, just like the 8 predecessors, may be valuable to analyze employees and scholars in any respect degrees who're attracted to the Fibonacci numbers and their functions. March sixteen, 2003 The Editor Fredric T. Howard arithmetic division Wake woodland collage field 7388 Reynolda Station Winston-Salem, NC 27109 xxi THE ORGANIZING COMMITTEES neighborhood COMMITTEE overseas COMMITTEE Calvin lengthy, Chairman A. F. Horadam (Australia), Co-Chair Terry Crites A. N. Philippou (Cyprus), Co-Chair Steven Wilson A. Adelberg (U. S. A. ) C. Cooper (U. S. A. ) Jeff Rushal H. Harborth (Germany) Y. Horibe (Japan) M. Bicknell-Johnson (U. S. A. ) P. Kiss (Hungary) J. Lahr (Luxembourg) G. M. Phillips (Scotland) J. 'Thrner (New Zealand) xxiii xxiv record OF CONTRlBUTORS TO THE convention * ADELBERG, ARNOLD, "Universal Bernoulli Polynomials and p-adic Congruences. " *AGRATINI, OCTAVIAN, "A Generalization of Durrmeyer-Type Polynomials. " BENJAMIN, paintings, "Mathemagics.

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**Example text**

Secondly we choose a = 0, b The same relations (4) imply (Bm,neo)(x) = n-l L k=m = 15m and this leads to the identity Bm,neo (n-m-1) xk(l - xt- k = k_m (1 - x)xm, if m = M~O,6m) eo. _ = 1, n -1, and (Bn,neo)(x) = xn. Substituting the above expressions of the function Am,neo, Bm,nel, m = 1, n, in the identity (8) we obtain the claimed result. 0 Remark: From (7) we deduce (Mneo)(O) = an and (Mneo)(l) = bn . This means that the convergence of (Mn)n~1 implies the convergence of the sequences a and b.

The median square, the third square, is located in cell 9. How many tilings contain exactly a dominoes to the left of the median square and exactly b dominoes to the right of the median square? Such a tiling has (a+b) dominoes and therefore (2n + 1) - 2(a + b) squares. Hence the median square has n - a - b squares on each side of it. Since the left side has (n - a - b) + a = n - b tiles, of which a are dominoes, there are (n~b) ways to tile to the left of the median square. Similarly, there are (n;B) ways to tile to the right of the median square.

Nk such that ni + 2n2 + ... + knk = n. Our tiling approach also succeeds in proving even more complex fibinomial identities. Identity 6: Question: In how many ways can we simultaneously tile n distinguishable boards of length + 1 with squares and dominoes? Answer 1: Since each board can be tiled fHI ways, there are fl:+1 such tilings. Answer 2: Condition on the number of dominoes covering each consecutive pair of cells. We claim there are (,::) (n~2al) ... (n-:;_,) ways to create n tilings of length k + 1 where al of them begin with dominoes, a2 have dominoes covering cells 2 and 3, and generally for 1 :S i :S k, ai of them have dominoes covering cells i and i + 1.