1,4

The polynomials demonstrate the Fibonacci product formula:

F(n) = PRODUCT_{k=1,(n-1)/2} (1 + 4*Cos^2(k*pi)/n).

Examples: n=7 relates to the heptagon. Product formula gives (4.24697,...),

(2.554958,...) and (1.19806222), product of these terms = 13 = F(7).

These are the roots to x^3 - 8x^2 - 19x - 13. Thus the product formula gives

the rightmost term of the polynomials and also the determinant of the

corresponding matrix, in this case = [2, -1, 0; -1, 3, -1; 0, -1, 3].

The second polynomial in the subset, x^3 - 9x^2 + 25x - 21; has

solutions/roots/e-vals through the product formula, polynomial and matrix

whose product = 21 and the determinant of the matrix = 21. The matrix in the

subset adds "1" to the position (1,1), thus: [3, -1, 0; -1, 3, -1, 0, -1, 3].

Row sums = A002530, denominators of continued fraction convergents to sqrt(3).

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]

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.

Triangle read by rows such that a pair has n terms, the first of which is the

characteristic polynomial for an (n-1) by (n-1) matrix of the form: (2,3,3,3,...) as the main

diagonal and (-1,-1,-1,..) as the sub and super diagonals.

Second of the subset pair has (3,3,3,...) as the main diagonal and (-1)'s in

the sub and super diagonals.

First few rows of the triangle are:

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, 55;

1, 14, 74, 183, 210, 89;

1, 15, 86, 234, 300, 144;

1, 17, 115, 394, 717, 654, 233;

1, 18, 130, 480, 951, 954, 377;

1, 20, 165, 725, 1825, 2622, 1985, 610;

1, 21, 183, 855, 2305, 3573, 2939, 987;

1, 23, 224, 1203, 3885, 7703, 9134, 5911, 1597;

1, 24, 245, 1386, 4740, 10008, 12707, 8850, 2584;

1, 26, 292, 1855, 7329, 18633, 30418, 30691, 17345, 4181;

1, 27, 316, 2100, 8715, 23373, 40426, 43398, 26195, 6765;

1, 29, 369, 2708, 12670, 39417, 82432, 114242, 100284, 50305, 10946;

1, 30, 396, 3024, 14770, 48132, 105805, 154668, 143682, 76500, 17711;

...

By row, alternate signs (+,-,+,-,...) with descending exponents. Rows with n

terms have exponents (n-1), (n-2), (n-3),...;

Example: There are two rows with 4 terms corresponding to the polynomials

x^3 - 8x^2 + 19x - 13 (roots associated with the heptagon); and

x^3 - 9x^2 + 25x - 21 (roots associated with the nonagon).

A000045, A002530

A125076 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 26 2008]

Sequence in context: A078657 A080959 A065548 this_sequence A022458 A084419 A119606

Adjacent sequences: A152060 A152061 A152062 this_sequence A152064 A152065 A152066

nonn,tabl

Gary W. Adamson & Roger L. Bagula (qntmpkt(AT)yahoo.com), Nov 22 2008