This site is supported by donations to The OEIS Foundation.

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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). 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.