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 + 5x, 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 + 5 x - D), c = 1/2 (1 + 5 x + D), where D = sqrt(5 + 6 x + 21 x^2).
EXAMPLE
First eight rows:
1
1 5
2 9 24
3 23 63 115
5 45 191 397 551
8 90 453 1381 2358 2640
13 170 1044 3807 9226 13482 12649
21 317 2249 9865 28785 58513 75061 60605
Row 4 represents the polynomial p(4,x) = 3 + 23 x + 63 x^2 + 115 x^3, so that (T(4,k)) = (3,23,63,115), k-0..3.
MATHEMATICA
p[1, x_] := 1; p[2, x_] := 1 + 5 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