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%I #5 Mar 30 2012 18:58:12
%S 1,4,5,8,26,27,30,31,42,45,46,120,121,124,125,136,137,140,141,184,187,
%T 188,199,200,203,204,502,503,506,507,518,519,522,523,566,567,570,571,
%U 582,583,586,587,758,761,762,773,774,777,778,821,822,825,826
%N Numbers that match even polynomials among the monic polynomials over {-1,0,1}, ordered as at A206821.
%C The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.
%e The first 13 polynomials:
%e 1 .... 1
%e 2 .... x
%e 3 .... x + 1
%e 4 .... x^2
%e 5 .... x^2 - 1
%e 6 .... x^2 - x
%e 7 .... x^2 - x - 1
%e 8 .... x^2 + 1
%e 9 .... x^2 + x
%e 10 ... x^2 + x + 1
%e 11 ... x^3
%e 12 ... x^3 - 1
%e 13 ... x^3 - x
%e Numbers n for which y(n,-x)=y(n,x): 1,4,5,8,26,...
%e Numbers n for which y(n,-x)=-y(n,x): 2,11,13,20,...
%t t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
%t b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
%t TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
%t f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
%t q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
%t q2[n_] := p[n - f[k] + 2]
%t y1 = Table[p[n], {n, 1, 4}];
%t Do[AppendTo[y1,
%t Join[Table[q1[n], {n, f[k], g[k] - 1}],
%t Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
%t y = Flatten[y1]; (* polynomials over {-1,0,1} *)
%t Flatten[Position[y - (y /. x -> -x), 0]] (* A207191 *)
%t Flatten[Position[y + (y /. x -> -x), 0]] (* A207192 *)
%Y Cf. A206821.
%K nonn
%O 1,2
%A _Clark Kimberling_, Feb 16 2012
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