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

1,2

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

Links

Table of n, a(n) for n=1..26.

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

Cf. A168193, A242364, A242365.

Sequence in context: A174013 A309266 A081056 * A066982 A246597 A179906

Adjacent sequences: A242445 A242446 A242447 * A242449 A242450 A242451

Keyword

nonn,easy,changed

Author

Clark Kimberling, Jun 11 2014

Status

approved