A118243 Triangle generated from Pell polynomials.

1, 1, 2, 1, 3, 5, 1, 4, 10, 12, 1, 5, 17, 33, 29, 1, 6, 26, 72, 109, 70, 1, 7, 37, 135, 305, 360, 169, 1, 8, 50, 228, 701, 1292, 1189, 408, 1, 9, 65, 357, 1405, 3640, 5473, 3927, 985, 1, 10, 82, 528, 2549, 8658, 18901, 23184, 12970, 2378, 1, 11, 101, 747, 4289

Offset

0,3

Comments

a(k)/a(k-1) of the array sequences tend to exp(arcsinh(N/2)) with rows starting N = 2, 3, 4, .... For example, terms of the Pell sequence row N=2 tend to converge to 2.414... = 1 + sqrt(2).

For scalar s, let V_m(s) be polynomials defined by V_0(s)=1, V_1(s)=s and V_m(s)=s*V_{m-1}(s)+V_{m-2}(s) (m>1). Then the generating array A (not the triangle) in the example below is given identically by the infinite matrix A=(A_{r,c}) with entries A_{r,c}=V_c(r+2); r=0,1,2,...; c=0,1,2,.... - L. Edson Jeffery, Aug 14 2011

Links

Table of n, a(n) for n=0..59.

Formula

Triangle, antidiagonals of the array in A073133, deleting the first row (Fibonacci numbers). Columns are generated as f(x) from the Pell polynomials (analogous to the Fibonacci polynomials).

Example

First few rows of the triangle:
  1;
  1, 2;
  1, 3,  5;
  1, 4, 10, 12;
  1, 5, 17, 33,  29;
  1, 6, 26, 72, 109, 70;
  ...
Deleting first row of the A073133 array, the generating array of the triangle is
  1, 2,  5,  12,  29, ...
  1, 3, 10,  33, 109, ...
  1, 4, 17,  72, 305, 1292, ...
  1, 5, 26, 135, 701, 3640, ...
  ...
By rows starting N = 2,3,... the generators of the array are a(k) = N(k-1)+ (k-2) (a generalized Fibonacci operation). Thus row (N=3) = 1, 3, 10, 33, ...
Columns of the array are generated from the terms of A038137 considered as Pell polynomials (analogous to the Fibonacci polynomials):
(1); (x + 1); (x^2 + 2x + 2); (x^3 + 3x^2 + 5x + 3); (x^4 + 4x^3 + 9x^2 + 10x + 5); and so on, where coefficient sums = the Pell numbers (1, 2, 5, 12, 29, ...).
k-th column of the triangle (offset T(0,0)) is generated from f(x), k-th degree Pell polynomial. For example, T(4,3)= 33, = f(2) using x^3 + 3x^2 + 5x + 3 = (8+12+10+3) = 33.

Crossrefs

Cf. A073133, A038137.

Sequence in context: A049069 A030237 A210557 * A210233 A347667 A297582

Adjacent sequences: A118240 A118241 A118242 * A118244 A118245 A118246

Keyword

nonn,tabl

Author

Gary W. Adamson, Apr 17 2006

Status

approved