|
| |
|
|
A162515
|
|
Triangle of coefficients of polynomials defined by Binet form: P(n,x) = (U^n-L^n)/d, where U=(x+d)/2, L=(x-d)/2, d=(4 + x^2)^(1/2).
|
|
3
|
|
|
|
0, 1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 1, 1, 0, 4, 0, 3, 0, 1, 0, 5, 0, 6, 0, 1, 1, 0, 6, 0, 10, 0, 4, 0, 1, 0, 7, 0, 15, 0, 10, 0, 1, 1, 0, 8, 0, 21, 0, 20, 0, 5, 0, 1, 0, 9, 0, 28, 0, 35, 0, 15, 0, 1, 1, 0, 10, 0, 36, 0, 56, 0, 35, 0, 6, 0, 1, 0, 11, 0, 45, 0, 84, 0, 70, 0, 21, 0, 1, 1, 0, 12, 0, 55
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,10
|
|
|
COMMENTS
|
Row sums 0,1,1,2,3,5,... are the Fibonacci numbers, A000045.
Note that the coefficients are given in decreasing order. - M. F. Hasler, Dec 07 2011
|
|
|
LINKS
|
Table of n, a(n) for n=0..96.
|
|
|
FORMULA
|
P(n,x)=x*P(n-1,x)+P(n-2,x), where P(0,x)=0 and P(1,x)=1.
|
|
|
EXAMPLE
|
First rows:
0 (row 0)
1
1...0
1...0...1
1...0...2...0
1...0...3...0...1
1...0...4...0...3...0
Row 6 matches P(6,x)=x^5 + 4*x^3 + 3*x.
|
|
|
MATHEMATICA
|
d = (4 + x^2)^(1/2); u = (x + d)/2; l = (x - d)/2;
f[n_] := (u^n - l^n)/d;
Factor[f[1]] (* P(1, x) *)
Factor[f[5]] (* P(5, x) *)
Factor[f[8]]
Factor[f[21]]
Factor[f[144]]
|
|
|
PROG
|
(PARI) P(n) = my( d=(4 + x^2)^(1/2), U=(x+d)/2, L=(x-d)/2); Pol((U^n-L^n)/d) \\ - M. F. Hasler, Dec 07 2011
|
|
|
CROSSREFS
|
Cf. A000045, A162514, A162516, A162517, A053119, A049310.
Sequence in context: A083280 A060689 A053119 * A175267 A108045 A143728
Adjacent sequences: A162512 A162513 A162514 * A162516 A162517 A162518
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
Clark Kimberling, Jul 05 2009
|
|
|
STATUS
|
approved
|
| |
|
|
