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A152063 Triangle read by rows, Fibonacci product polynomials. 9
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).

Row sums give A002530.

The triangle A125076 is formed by reading upward sloping diagonals. - Gary W. Adamson, Nov 26 2008

Bisection of the triangle: odd-indexed rows are reversals of the rows of A126124, even-indexed rows are the reversals of the rows of A123965. -  Gary W. Adamson_, Aug 15 2010

LINKS

Table of n, a(n) for n=1..68.

James P. Bradshaw, Philipp Lampe, Dusan Ziga, Snake graphs and their characteristic polynomials, arXiv:1910.11823 [math.CO], 2019. See 4.7 p. 16.

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.

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 * A336617 A321738 A022458

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 September 24 21:35 EDT 2020. Contains 337322 sequences. (Running on oeis4.)