A242448 Number of distinct linear polynomials b+c*x in row n of array generated as in Comments.

1, 3, 6, 12, 22, 38, 64, 106, 174, 284, 462, 750, 1216, 1970, 3190, 5164, 8358, 13526, 21888, 35418, 57310, 92732, 150046, 242782, 392832, 635618

Offset: 1

Author

Clark Kimberling, Jun 11 2014

Keywords

• nonn
• easy

Comments

Let f1(x) = 2x, f2(x) = 1-x, f3(x) = 2-x, g(1) = (x), and g(n) = union(f1(g(n-1)), f2(g(n-1)),f3(g(n-1))) for n >1. Let T be the array whose n-th row consists of the polynomials b + c*x arranged by the relation << defined by b1 + c1*x << b2 + c2*x if c1 < c2, and b1 + c1*x << b2 + c2*x if c1 = c2 and b1 < b2. If x = 1, the array is as at A242364.

Apparently a(n) = A168193(n-1) for 3 <= n <= 26. - Georg Fischer, Oct 23 2018

Examples

			First 3 rows of the array of linear polynomials:
x .......................................... (1 polynomial)
1-x ... 2-x ... 2x ......................... (3 polynomials)
1-2x .. 2-2x .. 4-2x .. -1+x .. 1+x .. 4x .. (6 polynomials)
		

Crossrefs

Cf. A168193, A242364, A242365.

Programs

Mathematica

z = 20; g[1] = {x}; f1[x_] := 2 x; f2[x_] := 1 - x; f3[x_] := 2 - x;
h[1] = g[1]; b[n_] := b[n] = Union[Expand[f1[g[n - 1]]], Expand[f2[g[n -
1]]], Expand[f3[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]];  u = Table[Length[g[n]], {n, 1, z}]  (* A242448 *)

Formula

Conjecture: a(n) = 2*a(n-1) - a(n-3) for n>= 6.