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A152063
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Triangle read by rows, Fibonacci product polynomials
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8
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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
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1,4
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COMMENTS
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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, Nov 26 2008]
Contribution 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)
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REFERENCES
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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.
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LINKS
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Table of n, a(n) for n=1..68.
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FORMULA
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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.
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EXAMPLE
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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).
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CROSSREFS
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A000045, A002530
A125076 [From Gary W. Adamson, Nov 26 2008]
Cf. A126124, A123965 [From Gary W. Adamson, Aug 15 2010]
Sequence in context: A078657 A080959 A065548 * A022458 A084419 A119606
Adjacent sequences: A152060 A152061 A152062 * A152064 A152065 A152066
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KEYWORD
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nonn,tabl
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AUTHOR
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Gary W. Adamson & Roger L. Bagula, Nov 22 2008
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STATUS
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approved
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