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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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. [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):216-221, 2004.

M. X He, D. Simon and P. E. Ricci, Dynamics of the zeros of Fibonacci polynomials, Fibonacci Quarterly, 35(2):160-168, 1997.

V. E. Hoggatt and C. T. Long, Divisibility Properties of Generalized Fibonacci Polynomials, Fibonacci Quarterly, 12:113-120, 1974.

FORMULA

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.

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 (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 9-gon (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

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Last modified August 15 18:49 EDT 2018. Contains 313779 sequences. (Running on oeis4.)