OFFSET
1,3
COMMENTS
Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.
LINKS
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
FORMULA
p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 3*x, u = p(2,x), and v = 1 - x - x^2.
p(n,x) = k*(b^n - c^n), where k = -(1/D), b = (1/2)*(1 + 3*x - D), c = (1/2)*(1 + 3*x + D), where D = sqrt(5 + 2*x + 5*x^2).
EXAMPLE
First ten rows:
1
1 3
2 5 8
3 13 19 21
5 25 59 65 55
8 50 137 231 210 144
13 94 316 623 834 654 377
21 175 677 1615 2545 2859 1985 987
34 319 1411 3859 7285 9691 9451 5911 2584
55 575 2849 8855 19115 30245 35105 30407 17345 6765
Row 4 represents the polynomial p(4,x) = 3 + 13*x + 19*x^2 + 21*x^3, so (T(4,k)) = (3,13,19,21), k=0..3.
MATHEMATICA
p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Nov 13 2023
STATUS
approved