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A207191
Numbers that match even polynomials among the monic polynomials over {-1,0,1}, ordered as at A206821.
3
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
1,2
COMMENTS
The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 16 2012
STATUS
approved