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^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 6*x + 5*x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(3*x + 1 + 1/k).
EXAMPLE
First eight rows:
1
1 3
2 6 8
3 15 25 21
5 30 76 90 55
8 60 188 324 300 144
13 114 439 948 1251 954 377
21 213 961 2529 4207 4527 2939 987
Row 4 represents the polynomial p(4,x) = 3 + 15*x + 25*x^2 + 21*x^3, so (T(4,k)) = (3,15,25,21), k=0..3.
MATHEMATICA
p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 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, Dec 25 2023
STATUS
approved