Search: id:A152063
Results 1-1 of 1 results found.

%I A152063
%S A152063 1,1,1,2,1,3,1,5,5,1,6,8,1,8,19,13,1,9,25,21,1,11,42,65,34,1,12,51,90,
%T A152063 55,1,14,74,183,210,89,1,15,86,234,300,144,1,17,115,394,717,654,233,6,
%U A152063 18,130,480,951,954,377,1,20,165,725,1825,2622,1985,610,1,21,183,855
%N A152063 Triangle read by rows, Fibonacci product polynomials
%C A152063 The polynomials demonstrate the Fibonacci product formula:
%C A152063 F(n) = PRODUCT_{k=1,(n-1)/2} (1 + 4*Cos^2(k*pi)/n).
%C A152063 Examples: n=7 relates to the heptagon. Product formula gives (4.24697,
               ...),
%C A152063 (2.554958,...) and (1.19806222), product of these terms = 13 = F(7).
%C A152063 These are the roots to x^3 - 8x^2 - 19x - 13. Thus the product formula 
               gives
%C A152063 the rightmost term of the polynomials and also the determinant of the
%C A152063 corresponding matrix, in this case = [2, -1, 0; -1, 3, -1; 0, -1, 3].
%C A152063 The second polynomial in the subset, x^3 - 9x^2 + 25x - 21; has
%C A152063 solutions/roots/e-vals through the product formula, polynomial and matrix
%C A152063 whose product = 21 and the determinant of the matrix = 21. The matrix 
               in the
%C A152063 subset adds "1" to the position (1,1), thus: [3, -1, 0; -1, 3, -1, 0, 
               -1, 3].
%C A152063 Row sums = A002530, denominators of continued fraction convergents to 
               sqrt(3).
%C A152063 A new triangle A125076 is formed by considering the A152063 rows as upward 
               sloping diagonals. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 26 2008]
%D A152063 N.D. Cahill and D.A. Narayan. "Fibonacci and Lucas Numbers as Tridiagonal 
               Matrix Determinants", Fibonacci Quaterly, 42(3):216-221, 2004 M.X 
               He, D. Simon and P.E. Ricci. "Dynamics of the zeros of Fibonacci 
               polynomials", Fibonacci Quaterly, 35(2):160-168, 1997. V.E. Hoggatt 
               and C.T. Long, "Divisibility Properties of Generalized Fibonacci 
               Polynomials". Fibonacci Quaterly, 12:113-120, 1974.
%F A152063 Triangle read by rows such that a pair has n terms, the first of which 
               is the
%F A152063 characteristic polynomial for an (n-1) by (n-1) matrix of the form: (2,
               3,3,3,...) as the main
%F A152063 diagonal and (-1,-1,-1,..) as the sub and super diagonals.
%F A152063 Second of the subset pair has (3,3,3,...) as the main diagonal and (-1)'s 
               in
%F A152063 the sub and super diagonals.
%e A152063 First few rows of the triangle are:
%e A152063 1;
%e A152063 1;
%e A152063 1, 2;
%e A152063 1, 3;
%e A152063 1, 5, 5;
%e A152063 1, 6, 8;
%e A152063 1, 8, 19, 13;
%e A152063 1, 9, 25, 21;
%e A152063 1, 11, 42, 65, 34;
%e A152063 1, 12, 51, 90, 55;
%e A152063 1, 14, 74, 183, 210, 89;
%e A152063 1, 15, 86, 234, 300, 144;
%e A152063 1, 17, 115, 394, 717, 654, 233;
%e A152063 1, 18, 130, 480, 951, 954, 377;
%e A152063 1, 20, 165, 725, 1825, 2622, 1985, 610;
%e A152063 1, 21, 183, 855, 2305, 3573, 2939, 987;
%e A152063 1, 23, 224, 1203, 3885, 7703, 9134, 5911, 1597;
%e A152063 1, 24, 245, 1386, 4740, 10008, 12707, 8850, 2584;
%e A152063 1, 26, 292, 1855, 7329, 18633, 30418, 30691, 17345, 4181;
%e A152063 1, 27, 316, 2100, 8715, 23373, 40426, 43398, 26195, 6765;
%e A152063 1, 29, 369, 2708, 12670, 39417, 82432, 114242, 100284, 50305, 10946;
%e A152063 1, 30, 396, 3024, 14770, 48132, 105805, 154668, 143682, 76500, 17711;
%e A152063 ...
%e A152063 By row, alternate signs (+,-,+,-,...) with descending exponents. Rows 
               with n
%e A152063 terms have exponents (n-1), (n-2), (n-3),...;
%e A152063 Example: There are two rows with 4 terms corresponding to the polynomials
%e A152063 x^3 - 8x^2 + 19x - 13 (roots associated with the heptagon); and
%e A152063 x^3 - 9x^2 + 25x - 21 (roots associated with the nonagon).
%Y A152063 A000045, A002530
%Y A152063 A125076 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 26 2008]
%Y A152063 Sequence in context: A078657 A080959 A065548 this_sequence A022458 A084419 
               A119606
%Y A152063 Adjacent sequences: A152060 A152061 A152062 this_sequence A152064 A152065 
               A152066
%K A152063 nonn,tabl
%O A152063 1,4
%A A152063 Gary W. Adamson & Roger L. Bagula (qntmpkt(AT)yahoo.com), Nov 22 2008

Search completed in 0.002 seconds