

A152063


Triangle read by rows, Fibonacci product polynomials


10



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, 6, 18, 130, 480, 951, 954, 377, 1, 20, 165, 725, 1825, 2622, 1985, 610, 1, 21, 183, 855
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OFFSET

1,4


COMMENTS

The polynomials demonstrate the Fibonacci product formula:
F(n) = PRODUCT_{k=1,(n1)/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/evals 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. [Gary W. Adamson, Nov 26 2008]
From Gary W. Adamson, Aug 15 2010: (Start)
Bisection of the triangle: odd indexed rows = reversals of A126124 rows.
Evens = reversals of A123965 rows. (End)


LINKS

Table of n, a(n) for n=1..68.
N. D. Cahill and D. A. Narayan, Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants, Fibonacci Quarterly, 42(3):216221, 2004.
M. X He, D. Simon and P. E. Ricci, Dynamics of the zeros of Fibonacci polynomials, Fibonacci Quarterly, 35(2):160168, 1997.
V. E. Hoggatt and C. T. Long, Divisibility Properties of Generalized Fibonacci Polynomials, Fibonacci Quarterly, 12:113120, 1974.


FORMULA

Triangle read by rows such that a pair has n terms, the first of which is the characteristic polynomial for an (n1) by (n1) 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.


EXAMPLE

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 (n1), (n2), (n3),...;
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 9gon (nonagon)).


CROSSREFS

Cf. A000045, A002530.
Cf. A125076. [Gary W. Adamson, Nov 26 2008]
Cf. A126124, A123965. [Gary W. Adamson, Aug 15 2010]
Sequence in context: A078657 A080959 A065548 * A022458 A084419 A119606
Adjacent sequences: A152060 A152061 A152062 * A152064 A152065 A152066


KEYWORD

nonn,tabf


AUTHOR

Gary W. Adamson & Roger L. Bagula, Nov 22 2008


STATUS

approved



