A152063 Triangle read by rows, Fibonacci product polynomials.

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

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 A341865 A347982

Adjacent sequences: A152060 A152061 A152062 * A152064 A152065 A152066

Keyword

nonn,tabf

Author

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

Status

approved