login
A242448
Number of distinct linear polynomials b+c*x in row n of array generated as in Comments.
5
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
1,2
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
FORMULA
Conjecture: a(n) = 2*a(n-1) - a(n-3) for n>= 6.
EXAMPLE
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)
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 11 2014
STATUS
approved