A207191 - OEIS

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

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OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..53.

EXAMPLE

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,...

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 *)

CROSSREFS

Cf. A206821.

Sequence in context: A104884 A226795 A113726 * A240790 A229861 A140315

Adjacent sequences: A207188 A207189 A207190 * A207192 A207193 A207194

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 16 2012

STATUS

approved