A368155 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 - 2*x^2.

1, 1, 3, 2, 3, 7, 3, 9, 5, 15, 5, 15, 26, 3, 31, 8, 30, 43, 63, -15, 63, 13, 54, 104, 87, 144, -81, 127, 21, 99, 203, 273, 115, 333, -275, 255, 34, 177, 416, 549, 609, -9, 806, -789, 511, 55, 315, 811, 1263, 1146, 1260, -725, 2043, -2071, 1023, 89, 555, 1573

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

Table of n, a(n) for n=1..58.

Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers 18 (2018) 1-28.

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 - 2*x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 - 6*x + x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

Example

First eight rows:
   1
   1    3
   2    3     7
   3    9     5    15
   5   15    26     3    31
   8   30    43    63   -15    63
  13   54   104    87   144   -81    127
  21   99   203   273   115   333   -275   255
Row 4 represents the polynomial p(4,x) = 3 + 9*x + 5*x^2 + 15*x^3, so (T(4,k)) = (3,9,5,15), k=0..3.

Mathematica

p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 3x - 2x^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

Cf. A000045 (column 1); A000225, (p(n,n-1)); A001787 (row sums), (p(n,1)); A002605 (alternating row sums), (p(n,-1)); A004254, (p(n,-2)); A057084, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368156.

Sequence in context: A007054 A377920 A084388 * A136389 A338032 A350770

Adjacent sequences: A368152 A368153 A368154 * A368156 A368157 A368158

Keyword

sign,tabl

Author

Clark Kimberling, Jan 20 2024

Status

approved