A368154 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3*x - x^2.

1, 1, 3, 2, 3, 8, 3, 9, 7, 21, 5, 15, 31, 15, 55, 8, 30, 53, 99, 30, 144, 13, 54, 124, 165, 306, 54, 377, 21, 99, 241, 447, 481, 927, 77, 987, 34, 177, 487, 909, 1509, 1341, 2767, 33, 2584, 55, 315, 941, 1995, 3135, 4905, 3605, 8163, -355, 6765, 89, 555

Offset: 1

Author

Clark Kimberling, Jan 20 2024

Keywords

• tabl
• sign

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.

Examples

			First eight rows:
   1
   1    3
   2    3     8
   3    9     7    21
   5   15    31    15    55
   8   30    53    99    30   144
  13   54   124   165   306    54  377
  21   99   241   447   481   927   77  987
		

Links

• Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), 1-28, 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 - 3*x - 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)*(2*x + 1 + 1/k).