cp's OEIS Frontend

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

%I A242448 #22 Apr 17 2024 16:41:39
%S A242448 1,3,6,12,22,38,64,106,174,284,462,750,1216,1970,3190,5164,8358,13526,
%T A242448 21888,35418,57310,92732,150046,242782,392832,635618
%N A242448 Number of distinct linear polynomials b+c*x in row n of array generated as in Comments.
%C A242448 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.
%C A242448 Apparently a(n) = A168193(n-1) for 3 <= n <= 26. - _Georg Fischer_, Oct 23 2018
%F A242448 Conjecture:  a(n) = 2*a(n-1) - a(n-3) for n>= 6.
%e A242448 First 3 rows of the array of linear polynomials:
%e A242448 x .......................................... (1 polynomial)
%e A242448 1-x ... 2-x ... 2x ......................... (3 polynomials)
%e A242448 1-2x .. 2-2x .. 4-2x .. -1+x .. 1+x .. 4x .. (6 polynomials)
%t A242448 z = 20; g[1] = {x}; f1[x_] := 2 x; f2[x_] := 1 - x; f3[x_] := 2 - x;
%t A242448 h[1] = g[1]; b[n_] := b[n] = Union[Expand[f1[g[n - 1]]], Expand[f2[g[n -
%t A242448 1]]], Expand[f3[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A242448 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]];  u = Table[Length[g[n]], {n, 1, z}]  (* A242448 *)
%Y A242448 Cf. A168193, A242364, A242365.
%K A242448 nonn,easy
%O A242448 1,2
%A A242448 _Clark Kimberling_, Jun 11 2014