A207191 Numbers that match even polynomials among the monic polynomials over {-1,0,1}, ordered as at A206821.

1, 4, 5, 8, 26, 27, 30, 31, 42, 45, 46, 120, 121, 124, 125, 136, 137, 140, 141, 184, 187, 188, 199, 200, 203, 204, 502, 503, 506, 507, 518, 519, 522, 523, 566, 567, 570, 571, 582, 583, 586, 587, 758, 761, 762, 773, 774, 777, 778, 821, 822, 825, 826

Offset: 1

Author

Clark Kimberling, Feb 16 2012

Keywords

• nonn

Comments

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

Examples

			The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
Numbers n for which y(n,-x)=y(n,x): 1,4,5,8,26,...
Numbers n for which y(n,-x)=-y(n,x): 2,11,13,20,...
		

Crossrefs

Cf. A206821.

Programs

Mathematica

t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,
  Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
Flatten[Position[y - (y /. x -> -x), 0]]  (* A207191 *)
Flatten[Position[y + (y /. x -> -x), 0]]  (* A207192  *)