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 + 4*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 + 4*x - D), c = (1/2)*(1 + 4*x + D), where D = sqrt(5 + 4*x + 12*x^2).
EXAMPLE
First nine rows:
1
1 4
2 7 15
3 18 38 56
5 35 116 186 209
8 70 273 650 859 780
13 132 629 1777 3366 3821 2911
21 246 1352 4600 10410 16556 16556 10864
34 449 2820 11024 29770 56874 78504 70356 405459
Row 4 represents the polynomial p(4,x) = 3 + 18*x + 38*x^2 + 56*x^3, so (T(4,k)) = (3,18,38,56), k=0..3.
MATHEMATICA
p[1, x_] := 1; p[2, x_] := 1 + 4 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